The 9 horizontal and 9 vertical lines on an 8 × 8 chess board form r rectangles (excluding squares) and s squares. The ratio s r in its lowest terms is:
In how many ways can 15 members of a council sit along a circular table, when the Secretary is to sit on one side of the Chairman and the Deputy Secretary on the other side?
If the letters of the word ‘SACHIN’ arranged in all possible ways and these words are written out as in dictionary, then the word SACHIN appears at serial number
The number of different words that can be formed with the letters of the word ‘MISSISSIPPI’, is
If n − 1 C r = k 2 − 3 n C r + 1 ,, then k belongs to
The number of injective functions from a set X containing m elements to a set Y containing n elements for m > n is:
The number of integral solutions of x + y + z = 0 with x ≥ − 5 , y ≥ − 5 , z ≥ − 5 is
If 1 a n P r + 1 = 1 b n P r = 1 c n P r − 1 then b 2 − ( a + b ) c is equal to
How many different words can be formed by using all the letters o( the word ‘ALLAHABAD’ such that vowels occupy the even positions, is
n C r + 2 n C r − 1 + n C r − 2 is equal to
The number of ways in which n distinct objects can be put into two identical boxes so that no box remains empty, is
If λ is the number of four digit numbers that can be formed by using the digits 1,2,3,4,5,6,7,8 and 9 such that the least digit used is 4,(Repetition of digits is allowed) then the value of λ 3 is
if three different dies are rolled ,then the number of ways in which the largest of three numbers is not 4 is equal to
Number of four-digit positive integers if the product of their digits is divisible by 3 is.
Let E = 1 3 + 1 50 + 1 3 + 2 50 + 1 3 + 3 50 + … + up to 50 terms, then the exponent of 2 in E! is
If 2 n + 1 P n − 1 : 2 n − 1 P n = 3 : 5 then the value of n is equal to
The number of words which can be formed out of the letters of the word ARTICLE , so that vowels occupy the even place is
If the letters of the word MOTHER are written in all possible orders and these words are written out as in a dictionary, then the rank of the word MOTHER is
In a circus there are ten cages for accommodating ten animals. Out of these four cages are so small that five out of 10 animals cannot enter into them. In how many ways will it be possible to accommodate ten animals in these ten cages?
The number of ways in which 10 candidates A 1 , A 2 , … , A 10 can be ranked such that A 1 , is always above A 10 is
The number of ways in which a team of eleven players can be selected from 22 players always including 2 of them and excluding 4 of them is
Eleven books consisting of 5 Mathematics, 4 Physics and 2 Chemistry are placed on a shelf. The number of possible ways of arranging them on the assumption that the books of the same subject are all together, is
A test consists of 6 multiple choice questions, each haying 4 alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is
Two straight lines intersect at a point O. Points A 1 , A 2 , …, A n are taken on one line and points B 1 , B 2 , …, B n on the other. If the point O is not to be used, the number of triangles that can be drawn using these points as vertices, is
The number of ordered pairs ( m , n ) , m , n ∈ { 1 , 2 , … , 100 } such that 7 m + 7 n is divisible by 5 is
If n + 5 p n + 1 = 1 2 ( 11 ) ( n − 1 ) n + 3 p n then n is equal to
The total number of permutations of n ( > 1 ) different things taken not more that r at a time, when a thing may be repeated any number of times, is
Sum of the series ∑ r = 1 n r 2 + 1 ( r ! ) is
The number of ways in which you can put five beads of five different colours to form a necklace is:
The total number of injective mappings from a set with m elements to a set with n elements for m > n , is
A group of 2n students consisting of n boys and n girls are to be arranged in a row such that adjacent members are of opposite sex. The number of ways in which this can be done is
Suppose p ∈ N , n = P C 2 x and m = n C 2 if 4 m : n = 18 : 1 , then p is equal to
The maximum possible number of points of intersection of 8 straight lines and 4 circles is
The number of solutions of 1 x + 1 y = 1 6 where x , y ∈ N
m men and w women are to be seated in a row so that no two women sit together. If m > w , then the number of ways in which they can be seated is:
The sum ∑ i = 0 m 10 i 20 m − i (where p q = 0 if p < q ) is maximum where m is
A library has n different books and 3 copies of each of the n books. The number of ways of selecting one or more books from the library is
The number of positive integers < 1,00,000 which contain exactly one 2, one 5 and one 7 in its decimal representation is
The number of 10 digit numbers that can be written by using the digits 0 and 1 is
If all the letters of the word ‘AGAIN’ be arranged as in a dictionary, then the.fiftieth word is
From 6 different novels and 3 different dictionaries, 4 novels and I dictionary are to be selected and arranged in a row on a self so that the dictionary is always in the middle. The number of such arrangement is
A student is to answer 10 out of 13 questions in an examination such that he must choose at least 4 from the first five questions. The number of choices available to him is
If a, b and c are the greatest values of C p 19 , C q 20 and C r 21 respectively, then: 11 C 9 19 a + b + c =
A student is to answer 10 out of 13 questions in an examination such that he must choose atleast 4 from the first five questions .The number of choices available to him is
20 passengers are to travel by a double decked bus which can accommodate 13 in the upper deck and 7 in the lower deck. The number of ways that they can be distributed if 5 refuse to sit in the upper deck and 8 refuse to sit in the lower deck is
There are 4n things of which n are alike and all the rest different. Then the number of permutations of 4n things taken 2n at a time, each permutation containing the n like things
Six people A,B,C,D,E,F are going to sit in a row on a bench. Let A and B are to be adjacent. C does not want to sit adjacent to D . E and F can sit anywhere. Number of ways in which these six people can be seated, is:
The number of ordered pairs ( m , n ) where m , n ∈ { 1 , 2 , 3 , … … 50 } such that 6 m + 9 n is a multiple of 5 is
The value of ∑ r = 2 10 r C 2 ⋅ 10 C r 10 is
The number of ways in which all the letters of the word “COCONUT” can be arranged such that at least one C comes at odd place, is
Two numbers are selected at random from a set of first 120 natural numbers, then the probability that product of selected number is divisible by 3 is
The number of ways in which 5 ladies and 7 gentlemen can be seated in a round table so that no two ladies sit together, is
The exponent of 3 in 100! is
How many even numbers of 3 different digits can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition of digits is not allowed)?
The value of 47 C 4 + ∑ r = 1 5 52 − r C 3 is equal to
If n C 3 + n C 4 > n + 1 C 3 then
A box contains two white balls, three black balls and four red balls. In how many ways can three balls be drawn from the box, if atleast one black ball is to be included in the draw?
If a polygon has 14 diagonals, then the number of its sides are
Sixteen men compete with one another in running, swimming and riding. How many prize lists could be made, if there were altogether 6 prizes of different values, one for running, 2 for swimming and 3 for riding?
The number of different seven digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is
If eight persons are to address a meeting then the number of ways in which a specified speaker is to speak before another specified speaker, is
If m = number of distinct rational numbers p q ∈ ( 0 , 1 ) such that p , q ∈ { 1 , 2 , 3 , 4 , 5 } and n = number of mappings from {1, 2, 3} onto {1, 2}, then m – n is
If a represents the number of permutations of (x + 2) things taken together, b represents the number of permutations of 11 things taken together out of x things, and c represents the number of permutations of (x – 11) things taken together so that a = 182bc, then x =
Suman writes letters to his five friends. The number of ways can be letters be placed in the envelopes so that atleast two of them are in the wrong envelopes are
The tensdigit of 1! + 2! + 3! + … + 49! is
A box contains two white balls, three black balls and four red balls. The number of ways in which three balls can be drawn from the box if atleast one black ball is to be included in the draw, is
The number of different 7 digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is
The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is
How many different nine digit numbers can be formed from the number 223355888 by rearranging its digits so that the odd digits occupy even positions?
In a certain test there are n questions. In this test 2 k students gave wrong answers to at least (n – k) questions, where k = 0, 1, 2,…, n. If the total number of wrong answers is 4095, then value of n is
For x ∈ R , let [x] denotes the greatest integer ≤ x, then the value of − 1 3 + − 1 3 − 1 100 + − 1 3 − 2 100 + , … , + − 1 3 − 99 100 is
Given five line segments of lengths 2, 3, 4, 5, 6 units. Then the number of triangles that can be formed by joining these lines is
In a network of railways, a small island has 15 stations. The number of different types of tickets to be printed for each class, if every station must have tickets for other station, is
On a new year day every student of a class sends a card to every other student. The postman delivers 600 cards. The number of students in the class are
In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. The number of ways in which the candidate can score 60% marks in aggregate is
The number of ways of choosing m coupons out of an unlimited number of coupons bearing the letters A, B and C so that they cannot be used to spell the word BAC, is
The number of non-negative integral solutions to the system of equations x + y + z + u + t = 20 and x + y + z = 5 is
In how many ways can 20 oranges be given to four children if each child should get at least one orange?
Eleven scientists are working on a secret project. They wish to lock up the documents in a cabinet such that cabinet can be opened if six or more scientists are present. Then, the smallest number of locks needed is
If a, b, c are three natural numbers in A.P. such that a + b + c = 21, then the possible number of values of a, b, c is
There are three piles of identical yellow, black and green balls and each pile contains at least 20 balls. The number of ways of selecting 20 balls if the number of black balls to be selected is twice the number of yellow balls, is
In a certain test there are n questions. In this test 2 k students gave wrong answers to at least ( n – k ) questions, where k = 0 , 1 , 2 , … , n . If the total number of wrong answers is 4095 , then value of n is
If m + n P 2 = 90 and m − n P 2 = 30 , then ( m , n ) is equal to
The least positive integral value of x for which 10 C x − 1 > 2 10 C x
If n P r = 2520 and n C r = 21 then n is is equal to:
The number of integers x , y , z , w such that x + y + z + w = 20 and x , y , z , w ≥ – 1 , is
A set B contain 2007 elements. Let C be the set con sisting of subsets of B which contain at most 1003 elements. The number of elements in C is
If n P r = 1680 and n C r = 70 then n is equal to
In a group of 8 girls, two girls are sisters. The number of ways in which the girls can sit in a row so that two sisters are not sitting together is
Suppose P is a set containing n distinct elements. Let S = { ( x , y , z ) ∣ x , y , z , ∈ , P and at least two of x , y , z are equal }. The number of elements in S is
The number of ways in which three girls and nine boys can be seated in two cars, each having numbered seats, 3 in the front and 4 at the back, is
Three straight lines l 1 , l 2 and l 3 are parallel and lie in the same plane. Five points are taken on each of l 1 , l 2 and l 3 . The maximum number of triangles which can be obtained with vertices at these points, is
A four digit number of distinct digits is formed by using the digits 2, 3, 4, 5, 6, 7, 8. The number of such numbers which are divisible by 25, is
The number of subsets of the set A = a 1 , a 2 , … , a n which contain even number of elements is
If 0 < r < s ≤ n and n P r = n P s then value of r + s is
Let a n = ∑ r = 0 n 1 n C r 2 and b n = ∑ r = 0 n ( n − r ) n C r 2 then b n equals
The number of diagonals of a polygon of 15 sides is
5-digit numbers are formed using 2, 3, 5, 7, 9 without repeating the digits. If p is the number of such numbers that exceeds 20000 and q be the number of those that lie between 30000 and 90000 then p : q is:
The number of ways of arranging 20 boys so that 3 particular boys are separated is
In a class of 10 students there are 3 girls students. The number of ways in which they can be arranged in a row, so that no two girls are consecutive is k ( 8 ! ) , then k is equal to
The range of the function f ( x ) = 15 − x P x − 8 is
Rakshit is allowed to select ( n + 1 ) or more books out of ( 2 n + 1 ) distinct books. If the number of ways in which he may not select all of them is 255, then value of n is
Let n = 2015 The least positive integer k for which k n 2 n 2 − 1 2 n 2 − 2 2 n 3 − 3 2 … n 2 − ( n − 1 ) 2 = r ! for some positive integer r is
Let T n denote the number of triangles which can be formed by using the vertices of a regular polygon of n sides. If T n + 1 − T n = 21 , then n equals
The value of E = ( 1 + 17 ) 1 + 17 2 1 + 17 3 ⦠1 + 17 19 ( 1 + 19 ) 1 + 19 2 1 + 19 3 ⦠1 + 19 17 is
A five digit number divisible by 6 is to be formed by using the digits 0, 1, 2, 3, 4 and 8 without repetition. The total number of ways in which this can be done is
An eight digit number divisible by 9 is to be formed by using 8 digits out of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 without replacement. The number of ways in which this can be done is
If P r stands for P r n then sum of the series 1 + P 1 + 2 P 2 + 3 P 3 + … + n P n is
The number of ways in which n distinct objects can be put into two different boxes, is
If 12 P r = 11 P 6 + 6 ⋅ 11 P 5 then r is equal to
The number of ways in which n distinct objects can be put into two different boxes so that no box remains empty, is
The value of ∑ r = 1 n n P r r ! , is
If 2 n + 1 P n − 1 : 2 n − 1 P n = 3 : 5 then equal to
The number of divisors of 240 which are of the form 4 m + 2 , is
In a bag there are 5 red, 3 white and 4 black balls. Four balls are drawn from the bag. The number of ways in which at most 3 reel balls are selected is
If n = m C 2 the value of n C 2 is given by
The number of different matrices that can be formed with elements 0, 1 , 2 or 3, each matrix having 4 elements, is
The 11 number of arrangements that can be made with the letters of the word ‘MATHEMATICS’ in which all vowels come together, is
The sum ∑ i = 0 m. 10 C i × 20 C m − i where p C q = 0 if p < q is maximum, when m is
The number of natural numbers less than 7 , 000 which can be formed by using the digits 0 , 1 , 3 , 7 , 9 (repetition is allowed) is equal to
If n − 1 C r = k 2 − 3 n C r + 1 , then k belongs to
The number 24 ! is divisible by
If a and b are the greatest values of 2 n C r and 2 n − 1 C r respectively. Then
The letters of the word COCHIN are permuted all the permutations arc arranged in an alphabetical order as in dictionary. The number of Words that appear before the word is COCHIN is
If n − 1 C 3 + n − 1 C 4 > n C 3 , then,
Total number of 6-digit numbers in which only one digit of 1,3,5,7 and 9 can be repeated and all the five digits 1, 3, 5, 7 and 9 appear is:
The number of ordered pairs (r,k) for which 6. 35 C r = k 2 − 3 . 36 C r + 1 , where k is an integer is:
If a, b and c are the greatest values of C p 19 , 20 C q and C r 21 respectively, then:
An urn contains 5 red marbles, 4 black marbles and 3 white marbles. Then the number of ways in which 4 marbles can be drawn so that at the most three of them are red is
If the number of five digit numbers with distinct digits and 2 at the 10th place is 336 k, then k is equal to:
The missing value in the following figure is
A dictionary is printed consisting of 7 lettered words only that can be made by rearranging the letters of the word CRICKET. If the words are printed in alphabetical order, as in a standard dictionary, then the number of words before the word CRICKET is
The number of five-letter words formed with the letters of the word CALCULUS is
A seven digit number is in the form of a b c defg ( g , f , e , etc. are digits at units, tens, hundreds place etc.) where a < b < c < d > e > f > g . Then the number of such possible numbers is
The total number of ways in which three distinct numbers in Arithmetic Progression can be selected from the set 1 , 2 , 3 , ……. , 24 is equal to
Let A = 1 10 , 1 9 , 1 8 , … … … , 1 3 , 1 2 , 1 , 2 , 3 , 4 , … … .8 , 9 , 10 , then the number of ordered pairs ( a , b ) such that a , b ∈ A and
Let the matrix A = x y − z 1 2 3 1 1 2 where x , y , z ∈ N . If ∣ a d j adj ( adj ( a d j A ) ) | | = 4 8 5 16 then number of such ( x , y , z ) are
Number of ways in which 7 green bottles and 8 blue bottles can be arranged in a row if exactly 1 pair of green bottles is side by side, is (Assume all bottles to be alike except for the colour):
The range of the function f x = 7 – x P x – 3 is
The number of ways of selecting 10 objects out 10 identical and 20 distinct objects is
Find the number of zeros (cyphers) at the end of 100!
The number of positive integral solutions of x 1 x 2 x 3 x 4 = 770 is
If x < 4 < y x , y ∈ { 1 , 2 , 3 , … , 10 } then find the number of ordered pairs (x, y)
There are 10 points in a plane, out of these 6 are collinear. lf N is the number of triangles formed by joining these points, then
Aiay writes letters to his five friends and addresses the corresponding. The number of ways can the letters be placed in the envelops so that atleast two of them are in the wrong envelopes are
How many different non-digit numbers can be formed from the digits of the number 223355888 by rearrangement of the digits so that the odd digits occupy even places?
We are to form different words with the letters of the word INTEGER. Let m 1 , be the number of words in which I and N are never together and m 2 be the number of words which begin with I and end with R, then m 1 /m 2 , is equal to
If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is
The total number of permutations of n (> 1) Different things taken not more than r at a time, when each thing may be repeated any number of times is
If eleven members of a committee sit at a round table so that the President and Secretary always sit together, then the number of arrangements is
The number of ways in which seven persons can be arranged at a round table, if two particular persons may not sit together is
20 persons are invited for a party. In how many different ways can they and the host be seated at circular table, if the two particular persona are to be seated on either side of the host?
In how many ways can 5 boys and 5 girls sit in a circle so that no two boys sit together?
How many numbers lying between 10 and 1000 can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition of digits is allowed) ?
How many 10-digit numbers can be written by using the digits 1 and 2 ?
The number of times the digit 3 will be written when listing the integers from 1 to 1000 is
The figures 4,5, 6, 7,8 are written in every possible order. The number of numbers greater than 56000 is
In how many ways can a student choose a program of 5 courses, if 9 courses are available and 2 specific courses are compulsory for every student?
Every body in a room shakes hands with everybody else. The total number of hand shakes is 66. The total number of persons in the room is
There are 10 lamps in a hall. Each one of them can be switched on independently. Find the number of ways in which hail can be illuminated.
In an examination, a student has to answer 4 questions out of 5 questions; questions 1 and 2 are however compulsory. Find the number of ways in which the student can make the choice.
In a football championship, there were played 153 matches. Every team played one match with each other. The number of team 8 participating in the championship is
A father with 8 children takes them 3 at a time to the zoological gardens, as often as he can without taking the same 3 children together more than once. The number of times he will go the garden, is
A car will hold 2 in the front seat and 1 in the rear seat. If among 6 persons 2 can drive, then number of ways in which the car can be filled, is
Six X’s have to be placed in the square of the figure such that each row contains atleast one ‘X’. In how many different ways can this be done?
A question paper is divided into two parts A and B and each part contains 5 questions. The number of ways in which a candidate can answer 6 questions selecting atleast two questions from each part is
Five balls of different colours are to be placed in three boxes of different sizes. Each box can hold all five balls. In how many ways can we place the balls so that no box remains empty?
The number of ways of dividing 52 cards amongst four players so that three players have 17 cards each and the fourth players just one card, is
In how many ways can Rs.16 be divided into 4 persons when none of them get less than Rs. 3?
Let f : { 1 , 2 , 3 , 4 , 5 } { 1 , 2 , 3 , 4 , 5 } that are onto and f ( i ) ≠ i is equal to
The number of triangles that are formed by choosing the vertices from a set of 12 points, seven of which lie on the same line is
The number of parallelograms that can be formed from a set of four parallel lines intersecting another set of three parallel lines is
The maximum number of points of intersection of 6 circles is
The number of diagonals in a polygon of m sides is
The number of triangles that can be formed by 5 points in a line and 3 points on a parallel Iine is
The greatest possible number of points of intersection of 8 straight lines and 4 circles is
Let A be the set of 4-digit numbers a 1 a 2 a 3 a 4 where a 1 < a 2 < a 3 < a 4 then n (A) is equal to
The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ‘SYLLABUS’ such that two letters are distinct and two letters are alike.is …… .
The letters of the word RANDOM are written in all possible orders and these words are written out as in a dictionary then the rank of the word RANDOM is
Let A = {1, 2, 3, 4} and B = {1, 2}. Then, the number of onto functions from A to B is:
The number of ways of selecting 10 balls from the unlimited number of red, green, white and yellow balls, if selection must include 2 red and 3 yellow balls, is
The number of permutations of letters a, b, c, d, e, f, g so that neither the pattern beg nor cad appears is
How many different nine digit numbers can be formed from the number 22 33 55 8 88 by rearranging its digits so that the odd digits occupy even positions?
For a game in which two partners play against two other partners, six persons are available. If every possible pair must play with every other possible pair, then the total number of games played is
Every body in a room shakes hands with every body else. The total number of hand shakes is 66. The total number of persons in the room is
If the number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232. Then n is equal to
m C r + 1 + = ∑ k = m n k C r =
The number of divisors a number 38808 can have, excluding 1 and the number itself is
If the letters of the word MOTHER are written in all possible orders and these words are written out as in a dictionary, then the rank of the word MOTHER is
The number of positive integral solutions of 15 < x 1 + x 2 + x 3 ≤ 20 , is equal to
Let S be the set of all functions from the set A to the set A. If n(A) = k then n(S) is
There are three coplanar parallel lines. If any p points are taken on each of the lines, the maximum umber of triangles with vertices at these points is
The number of ways in which thirty five apples can be distributed among 3 boys so that each can have any number of apples, is
The number of non-negative solutions of x 1 + x 2 + x 3 + , … , + x n ≤ n (where n is positive integer) is
There are n concurrent lines and another line parallel to one of them. The number of different triangles that will be formed by the (n + 1) lines, is
The number of ordered pairs (m, n), m, n ∈ { 1 , 2 , … 50 } such that 6 n + 9 m is a multiple of 5 is
If n is even and n C 0 < n C 1 < n C 2 < … < n C r > n C r + 1 > … > n C n then r =
A set contains (2n + 1) elements. The number of subsets of the set which contain at most n elements
An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is
If 20% of three subsets (i.e., subsets containing exactly three elements) of the set A = {a 1 , a 2 ,…, a n } contain a1, then the value of n is
Eleven animals of a circus have to be placed in eleven cages one in each cage. If 4 of the cages are too small for 6 of the animals, then the number of ways of caging the animals i
The number of ways of choosing n objects out of (3n + 1) objects of which n are identical and (2n + 1) are distinct, is
A crocodile is known to have not more than 68 teeth. The total number of crocodiles with different set of teeth is
The number of two digit numbers which are of the form xy with y < x are given by
If all permutations of the letters of the word AGAIN are arranged as in dictionary, the forty ninth word is
The total number of ways in which a beggar can be given at least one rupee from four 25 p. coins, three 50 p. coins and 2 one rupee coins is
A student is allowed to select atmost n books from a collection of (2n + 1) books. If the total number of ways in which he can select books is 63, then n =
The number of permutations of the letters a, b, c, d such that b does not follow a, c does not follow b, and d does not follow c, is
If S = ∑ r = 0 m n + r C k , then
The number of ways of dividing 15 men and 15 women into 15 couples, each consisting of a man and a woman, is
Statement 1: The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is 9 C 3 . Statement 2: The number of ways of choosing any 3 places from 9 different places is 9 C 3 .
If the number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232. Then n is equal to
Assuming the balls to be identical except for difference in colours, the number of ways in which one or more balls can be selected from 10 white, 9 green and 7 black balls is
The number of 4-digit numbers with distinct digits is
How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent?
In a shop there are five types of ice-creams available. A child buys six ice-creams. Statement 1: The number of different ways the child can buy the six ice-creams is 10 C 5 . Statement 2: The number of different ways the child can buy the six ice-creams is equal to the number of different ways of arranging 6 A’s and 4 B’s in a row.
From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is
In a certain test, a i students gave wrong answers to at least i questions where i = 1, 2, 3, …, k. No student gave more than k wrong answers. The total number of wrong answers given is
A gentleman invites 13 guests to a dinner and places 8 of them at one table and remaining 5 at the other, the tables being round. The number of ways he can arrange the guests is
There are stalls for 10 animals in a ship. The number of ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 10, is
∑ ∑ 0 ≤ i ≤ j ≤ 10 10 C j j C i is equal to
If eight persons are to address a meeting then the number of ways in which a specified speaker is to speak before another specified speaker, is
The number of divisors a number 38808 can have, excluding 1 and the number itself is
In an examination a candidate has to pass in each of the papers to be successful. If the total number of ways to fail is 63, how many papers are there in the examination?
The sum of all the numbers that can be formed with the digits 2, 3, 4, 5 taken all at a time is
The number of different 7 digit numbers that can be written using only the three digits 1, 2 and 3 with the condition that the digit 2 occurs twice in each number is
The number of ways of choosing n objects out of (3n + 1) objects of which n are identical and (2n + 1) are distinct, is
There are 10 points in a plane of which no three points are collinear and 4 points are concyclic. The number of different circles that can be drawn through at least 3 of these points is
The number of permutations of letters a, b, c, d, e, f, g so that neither the pattern beg nor cad appears is
For x ∈ R, let [x] denotes the greatest integer ≤ x, then the value of − 1 3 + − 1 3 − 1 100 + − 1 3 − 2 100 + … + − 1 3 − 99 100 is
A set contains (2n + 1) elements. The number of subsets of the set which contains at most n elements is
If 20% of three subsets (i.e., subsets containing exactly three elements) of the set A = {a 1 , a 2 , …, a n } contain a1, then the value of n is
If all permutations of the letters of the word AGAIN are arranged as in dictionary, the forty-ninth word is
Let y be an element of the set A = {1, 2, 3, 5, 6, 10, 15, 30} and x 1 , x 2 , x 3 be integers such that x 1 x 2 x 3 = y, then the number of positive integral solutions of x 1 x 2 x 3 = y is
In a certain test there are n questions. In this test 2 k students gave wrong answers to at least (n – k) questions, where k = 0, 1, 2, …, n. If the total number of wrong answers is 4095, then value of n is
If the number of ways in which n different things can be distributed among n persons so that at least one person does not get any thing is 232. Then, n is equal to
If S = ∑ r = 0 m n + r C k , then
Given 5 different green dyes, 4 different blue dyes and 3 different red dyes, the number of combinations of dyes that can be chosen by taking at least one green and one blue dye is
If m = number of distinct rational numbers p q ∈ ( 0 , 1 ) such that p, q ∈ {1, 2, 3, 4, 5} and n = number of mappings from {1, 2, 3} onto {1, 2}, then m – n is
Number of points having position vector a i ^ + b j ^ + c k ^ , where a, b, c ∈ {1, 2, 3, 4, 5} such that 2 a + 3 b + 5 c is divisible by 4 is
2 n C r ( 0 ≤ r ≤ 2 n ) is greatest when r is equal to
The number of even numbers greater than 100 that can be formed by the digits 0, 1, 2, 3 (no digit being repeated) is
In a certain city, all telephone numbers have six digits, the first two digits always being 41 or 42 or 46 or 62 or 64. The number of telephone numbers having all six digits distinct is
The number of positive numbers less than 1000 and divisible by 5 (no digit being repeated) is
The total number of ways of selecting five letters from the letters of the word INDEPENDENT, is
The sum of five digit numbers which can be formed with the digits 3, 4, 5, 6, 7 using each digit only once in each arrangement, is
The sum of all the numbers that can be formed by writing all the digits 3, 2, 3, 4 only once is
The sum of all numbers greater than 1000 formed by using the digits 0, 1, 2, 3, no digit being repeated in any number, is
The sum of all numbers greater than 1000 formed by using the digits 0, 1, 2, 3, no digit being repeated in any number, is
The number of four digit numbers that can be formed from the digits 0, 1, 2, 3, 4, 5 with at least one digit repeated is
A table has provision for 7 seats, 4 being on one side facing the window and 3 being on the opposite side. The number of ways in which 7 people can be seated at the table if 2 people, X and Y, must sit on the same side, is
The number of odd numbers lying between 40000 and 70000 that can be made from the digits 0, 1, 2, 4, 5, 7 if digits can be repeated in the same number is
There are four oranges, five apples and six mangoes in a fruit basket. The number of ways in which a person can make a selection of fruits among the fruits in the basket, is
The largest integer n such that 33! is divisible by 2 n is
The product of r consecutive positive integers is divisible by
The number of ordered triplets of positive integers which are solutions of the equation x + y + z = 100 is
The number of non-negative integral solutions of x 1 + x 2 + x 3 + 4x 4 = 20 is
The number of words that can be formed, with the letters of the work ‘Pataliputra’ without changing the relative order of the vowels and consonants, is
The number of zeros at the end of 100! is
The number of integers between 1 and 1000000 that have the sum of the digits 18, is
The number of ways in which 16 identical things can be distributed among 4 persons if each person gets at least 3 things, is
The number of ways in which 30 marks can be alloted to 8 questions if each question carries at least 2 marks, is
If ‘n’ is an integer between 0 and 21, then the minimum value of n! (21 – n)! is
The number of 7 digit numbers the sum of whose digits is even, is
The number of numbers greater than 10 6 that can be formed using the digits of the number 2334203, if all the digits of the given number must be used, is
The number of positive integral solutions of the inequality 3x + y + z ≤ 30, is
A train is going from Delhi to Indore, stops at nine intermediate stations. Six persons enter the train during the journey with six different tickets. The number of different sets of tickets possessed by them is
An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only the three digits 2, 5 and 7. The smallest value of n for which this is possible is
The total number of 5-digit numbers of different digits in which the digit in the middle is the largest is
The number of ways in which a mixed doubles game can be arranged from amongst n couples such that no husband and wife play in the same game, is
Six X’s have to be placed in squares of the figure given below, such that each row contains at least one X. The number of different ways in which this can be done is
If n + 2 C 3 = n + 3 P 2 − 20 then n is equal to:
An n-digit number is a positive number with exactly n digits. Nine hundred distinct n-digit numbers are to be formed using only three digits 2 , 5 and 7 . The smallest value of n for which this is possible is
Let Q n be the number of possible quadrilaterals formed by joining vertices of an n sided regular polygon. If Q n + 1 − Q n = 20 then value of n is:
The number of ways of arranging p numbers out of 1 , 2 , 3 , … , q so that maximum is q – 2 and minimum is 2 (repetition of number is allowed) such that maximum and minimum both occur exactly once , ( p > 5 , q > 3 ) is
If there are 62 onto mapping from a set X containing n elements to the set Y = { − 1 , 1 } , then n is equal to
The number of ways of selecting 4 cards of an ordinary pack of playing cards so that exactly 3 of them are of the same denomination is
The maximum possible number of points of intersection of 7 straight lines and 5 circles is:
If 20 % of three subsets (i.e., subsets containing exactly three elements) of the set A = a 1 , a 2 , … , a n contain a 1 then value of n is
The exponent of 7 in the prime factorization of 100 C 50 is
If 15 C r + 1 : 15 C 3 r = 3 : 11 then value of r is
If there are 30 onto, mappings from a set containing n elements to the set 0 , 1 , then n equals:
Let A be a set containing ten elements. Then the number of subsets of A containing at least four elements is
If x ∈ N and ( 2 x ) ! 3 ! ( 2 x − 3 ) ! : x ! 2 ! ( x − 2 ) = 44 : 3 , then x is equal to
If n C 4 , n C 5 and n C 6 are in A.P., then a value of n can be
If x ∈ N and ( x + 2 ) ! ( 2 x − 1 ) ! ⋅ ( 2 x + 1 ) ! ( x + 3 ) ! = 72 7 , then x is equal to
The least positive integer n for which n − 1 C 5 + n − 1 C 6 < n C 7 is
A committee consisting of at least three members is to be formed from a group of 6 boys and 6 girls such that it always has a boy and a girl. The number of ways to form such committee is:
The number of ways of permuting letters of the word E N D E A N O E L so that none of the letters D , L , N occurs in the last five positions is
If 2 n C 4 , 2 n C 5 and 2 n C 6 are in A.P, then n is equal to
If letters of the word S A C H I N are arranged in all possible ways and are written out as in a dictionary, then the word S A C H I N appears at serial number
If n + 1 C r + 1 : n C r : n − 1 C r − 1 = 11 : 6 : 3 then the value of n is
The set S = { 1 , 2 , 3 , … 12 } is to be partitioned into three sets A , B , C of equal size. Thus A ∪ B ∪ C = S , A ∩ B = B ∩ C = C ∩ A = ϕ The number of ways to partition S is
Suppose that six students, including Madhu and Puja, are having six beds arranged in a row. Further, suppose that Madhu does not want a bed adjacent to Puja. Then the number of ways, the beds can be allotted to students is
Ten letters of an alphabet are given. Words with five letters are formed by these given letters. Then the number of words which have at least one letter repeated is
The number of words that can be formed by using the letters of the word MATHEMATICS that start as well as end with T is
In a car with seating capacity of exactly five persons, two persons can occupy the front seat and three persons can occupy the back seat. If amongst the seven persons, who wish to travel by this car, only two of them know driving then number of ways in which the car can be fully occupied and driven by them, is:
If a n = ∑ r = 0 n 1 n C r , then ∑ r = 0 n r n C r equals:
If S = ∑ i = 1 n n C i − 1 n C i + n C i − 1 3 = 36 13 then n is equal to
A code word of length 4 consists two distinct consonants in the English alphabet followed by two digits from 1 to 9, with repetition allowed in digits. If the number of code words so formed ending with an even digit is 432 k , then k is equal to
A man invites 6 non-vegeterian and 5 vegeterian friends for a dinner party. He arrange 6 non-vegeterian friends on one round table and 5 vegeterian friends along another round table. The number of ways this can be done is:
A library has n different books and has p copies of each of the book. The number of ways of selecting one or more books from the library is
In an examination, there are 11 papers. A candidate has to pass in at least 6 papers to pass the examination. The number of ways in which the candidate can pass the examination is:
The number of ways in which we can get a score of 11 by throwing three dice is
At an election, a voter may vote for any number of candidates not greater than the number to be chosen. There are 10 candidates and 5 members to be selected. The number of ways of in which a voter can vote is:
The number of 9 digit numbers which have all distinct digits, is
The number of ways of putting 5 identical balls in 10 identical boxes, if not more than one can go in a box is
The number of ways of factoring 91,000 into two factors m and n such that m > 1 , n > 1 and g c d ( m , n ) = 1 is
The number of four digit numbers that do not contain 4 different digits is:
The sum S = 1 9 ! + 1 3 ! 7 ! + 1 5 ! 5 ! + 1 7 ! 3 ! + 1 9 ! equals
The number of ways of dividing 10 girls into two groups of 5 each so that two shortest girls are in the different groups is:
Let a n = 10 n n ! for r n ≥ 1 . Then a n take the greatest value when n equals
Let m and n be two digit natural numbers. The number of pairs ( m , n ) such that n can be subtracted from m without borrowing is:
If E = 1 4 ⋅ 2 6 ⋅ 3 8 ⋅ 4 10 ⋯ 30 62 ⋅ 31 64 = 8 x , then value of x is
A five digit number divisible by 3 is to be formed using the digits 0, 1, 2, 3, 4 and 5 without repetition. The total number of ways, this can be done is
The number of subsets of a set containing n distinct object is
Kunal Gaba has n objects, each of weight w . He weighs them in pairs and finds the sum of the weights of all possible pairs is 120g. When his friend Rakshit weighs them in triplets, the sum of all possible weights is 240g. The value of n is
If a n = ∑ r = 0 n 1 n C r , then value of ∑ r = 0 n n − 2 r n C r is
The number of rational numbers lying in the interval (2015, 2016) all whose digits after the decimal point are non-zero and are in decreasing order is
A committee of at least three members is to be formed from a group of 6 boys and 6 girls such that it always has a boy and a girl, The number of ways to form such a committee is
The number of ways in which a committee of 3 women and 4 men be chosen from 8 women and 7 men if Mr. X refuses to serve on the committee if Mr. Y is a member of the committee is
There are 10 points in a plane, out of these 6 are collinear. If N is the number of triangles formed by joining these points, then
The product of first n odd natural numbers equals
Assuming the balls to be identical except for difference of colours the number of ways in which one or more ball can be selected from 10 white, 9 green and 7 black balls is:
The number of functions f from the set A = { 0 , 1 , 2 } in to the set B = { 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 } such that f ( i ) ≤ f ( j ) for i < j and i , j ∈ A is
2 m white counters and 2 n red counters are arranged in a straight line with ( m + n ) counters on each side of a central mark. The number of ways of arranging the counters, so that the arrangements are symmetrical with respect to the central mark, is
The number of ways in which two teams A and B of 11 players each can be made up from 22 players so that two particular players are on the opposite sides is
The number of positive integral solution of the equation x 1 x 2 x 3 x 4 x 5 = 1050 is
The number of ways in which 20 letters x 1 , x 2 , … x 10 ; y 1 , y 2 , … , y 10 be arranged in a line so that suffixes of the letters x and also those of y are respectively in ascending order of magnitude is
There are 3 set of parallel lines containing respectively p lines, q lines and r lines respectively. The greatest number of parallelograms that can be formed by the system
The number of six digit numbers which have sum of their digits as an odd integer, is
The number of ways in which we can arrange the digits 1, 2, 3, …, 9 such that the product of five digits at any of the five consecutive positions is divisible by 7 is
The product of n consecutive natural number is always divisible by
The number of natural numbers less than one million that can be formed by using the digits, 0 , 2 , and 3 is
There are three pigeon holes marked M , P , C . The number of ways in which we can put 12 letters so that 6 of them are in M , 4 are in P and 2 are in C is
The number of integral solutions of x 2 − y 2 = 352706 is
Let X be a set containing n elements. The number of reflexive relations that can be defined on X is
The greatest number of points of intersection of n circles and m straight lines is
The number of binary sequences of length n that contain even number of 1’s is
The number of natural numbers with distinct digits is
Let A be a set containing n elements. The number of ways of choosing two subsets P and Q of A such that P ∩ Q = ϕ is
Suppose X contains m elements and Y contain n elements. The number of functions from X to Y is
The last digit of ( 1 ! + 2 ! + . . . . . . . + 2009 ! ) 500 is
Three boys and three girls are to be seated around a circular table. Among them the boy X does not want any girl as a neighbour and girl Y does not want any boy as a neighbour. The number of possible arrangement is:
The number of five digit numbers that contain 7 exactly once is
If 1 4 C n = 1 5 C n + 1 6 C n , then value of n is
If n C r : n C r + 1 : n C r + 2 = 1 : 2 : 3 then r is equal to
If n = 2 100 3 2 and d 1 , d 2 , … d k is the set of all divisors of n , then ∑ j = 1 k 1 d j equals
At an election there are five candidates and three members are to be elected, and a voter may vote for any number of candidates not greater than the number to be elected. The number of ways in which the person can vote is
The number of ways selecting three numbers from 1 to 30, so as to exclude every selection of three consecutive number is:
If all permutations of the letters of the word P E N C I L are arranged as in a dictionary, then 413th word is
If n is odd and n C 0 < n C 1 < n C 2 < … < n C r , then maximum possible value of r is
Out of 10 consonants and 4 vowels, words with 6 consonants and 3 vowels are formed. The number of such words is
If in the expansion of x 3 − 1 x 2 n the sum of the coefficients of x 5 and x 10 is 0 then the coefficient of the third term is:
The number of arrangements of the letters of the word BANANA in which two N’s do not appear adjacently is
Let S = 18 a b 25 : a , b ∈ N The number of singular matrices in s is
The number of ways we can put 5 different balls in 5 different boxes such that at most three boxes is empty, is equal to
The range of the function 7 − x P x − 3 is
The value of 50 C 4 + ∑ r = 1 6 56 − r C 3 is
The number of positive integers n such that 2 n di vides n! is
The number of five digit numbers that can be formed by using digits 1, 2, 3 only, such that three digits of the formed number are identical, is
The letters of the word C O C H I N are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before C O C H I N is
The number of ordered pairs ( m , n ) , m , n ∈ { 1 , 2 … , 50 } such that 6 m + 9 n is a multiple of 5 is
A class consists of 4 boys and g girls. Every Sunday five students, including at least three boys go for a picnic to Appu Ghar, a different group being sent every week. During, the picnic, the class teacher gives each girl in the group a doll. If the total number of dolls distributed was 85, then value of g is
The number of ordered pairs of integers ( x , y ) satisfying the equation x 2 + 6 x + y 2 = 4 is
The number of the factors of 20 ! is
In a football championship, 153 matches were played. Every team played one match with each other, the number of teams participating in the championship is
Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. The number of words which have at least one of their letter repeated is
The number of squares that we can find on a chess board is
The total number of ways of dividing m n distinct objects into n equal groups is
There are 15 points in a plane, no two of which are in a straight line except 4, all of which are in a straight line. The number of triangles that can be formed by using these 15 points is
The number of ways in which we can get at least 6 successive tails when a coin is tossed 10 times in a row, is
The greatest common divisor 31 C 3 , 31 C 5 … , 31 C 29 is
The number of ways of choosing n objects out of ( 3 n + 1 ) objects of which n are identical and ( 2 n + 1 ) are distinct, is
if n C 3 + n C 4 > n + 1 C 3 ,then
Let A be a set of n ( ≥ 3 ) distinct elements. The number of triplets (x, y, z) of the A elements in which at least two coordinates is equal to
In a room, there are 12 bulbs of the same wattage, each having a separate switch. The number of ways to light the room with different amount of illumination is
The number of five-digit numbers that contain 7 exactly once is
Numbers greater than 1000 but not greater than 4000 which can be formed with the digits 0, 1,2,3,4 (repetition of digits is allowed) are
The number of nine nonzero digits such that all the digits in the first four places are less than the digit in the middle and all the digits in the last four places are greater than that in the middle is
The total number of six-digit natural numbers that can be made with the digits 1 ,2, 3, 4, if all digits are to appear in the same number at least once is
If all the permutations of the letters in the word OBJECT are arranged (and numbered serially) in alphabetical order as in a dictionary, then the 717th word is
The number of words of four letters containing equal number of vowels and consonants, where repetition is allowed, is
Three boys of class X, four boys of class XI, and five boys of class XII sit in a row. The total number of ways in which these boys can sit so that all the boys of same class sit together is equal to
If n C r − 1 = 36 , n C r = 84 and n C r + 1 = 126 , then the value of r is equal to
