Important Binomial Theorem Questions for CBSE Class 11th

If C 0 , C 1 , C 2 , … , C n denote the binomial coefficients i11 the expansion of ( 1 + x ) n then a C 0 + ( a + b ) C 1 + ( a + 2 b ) C 2 + … + ( a + n b ) C n =

The sum of the last 30coefficients of power sof x the binomial expansion of ( 1 + x ) 59 is

Coefficient of x − 4 in ⁡ 3 2 − 3 x 2 10 is

The sum of the series S = 1 19 ! + 1 3 ! 17 ! + 1 5 ! 15 ! + … … … t o 10 terms is equal to

If O be the sum of odd terms and E that of even terms in the expansion of ( x + a ) n , then O 2 − E 2 =

If the coefficientsof x 7 and x 8 in 2 + x 3 n are equal, then n is

The coefficient of x 5 in the expansion of ( 1 + x ) 21 + ( 1 + x ) 22 + … + ( 1 + x ) 30 is

The coefficient of x 18 in the product ( 1 + x ) ( 1 − x ) 10 1 + x + x 2 9 is

If ( 1 + x ) n = C 0 + C 1 x + C 2 x 2 + … + C n x n them the value of C 0 + 2 C 1 + 3 C 2 + … + ( n + 1 ) C n will be

If A and B denote the coefficients of x n in the binomial expansions of ( 1 + x ) 2 n and ( 1 + x ) 2 n − 1 respectively, then

The coefficient of x 4 in the expansion of x 2 − 3 x 2 10 , is

If C 0 , C 1 , C 2 , C 3 , … , C n denote the binomial coefficients in the binomial expansion of ( 1 + x ) n then 1. C 1 − 2 ⋅ C 2 + 3 ⋅ C 3 − 4 ⋅ C 4 + … + ( − 1 ) n − 1 n C n =

T h e c o e f f i c e i n t o f t 24 i n t h e e x p a n s i o n o f ( 1 + t 2 ) 12 1 + t 12 1 + t 24 i s

The coefficient of x 50 in ⁡ ( 1 + x ) 41 1 − x + x 2 40 is

The constant term in the expansion of x 2 + 1 x 2 + y + 1 y 8 is

The value of 2008 3 – 2 2008 4 + 3 2008 5 – 4 2008 6 + . . . . . . . 2005 2008 2007 i s , w h e r e n r = C n , r , is

Coefficient of x 6 in ( 1 + x ) 1 + x 2 2 1 + x 3 3 …….. 1 + x n n , n ≥ 6 , is

If 7 103 is divided by 25 , then the remainder is

The sum of last two digits of the number 7 101 is

If the middle term of 1 x + x sin ⁡ x 10 is equal to 7 7 8 , then the value of x is

If the third term in the expansion of x + x log 10 x 5 is equal to 10 , 00 , 000 , then x =

The coefficient of x 9 in the expansion of (1 + x) (1 + x 2 ) (1 + x 3 ) … (1 + x 100 ) is .

In the expansion of the following expression 1 + ( 1 + x ) + ( 1 + x ) 2 + … + ( 1 + x ) n the coefficient of x k ( 0 ≤ k ≤ n ) is

The coefficient of X 53 in the following expansion ∑ m = 0 100 100 C m ( x − 3 ) 100 − m ⋅ 2 m is

If for positive integers r > 1 , n > 2 , the coefficient of the (3r)th and (r + 2)th powers of X in the expansion of (1 + x) 2n are equal, then

If the coefficients of three consecutive terms in the expansion of (1+x) n are in the ratio 1:7:42 then the value of n is

If the coefficients of P th, ( p + 1)th and ( p + 2)th terms in the expansion of (1 + x) n are in AP, then

The term independent of x in ( 1 + x ) m 1 + 1 x n , is

If the last term in the binomial expansion of 2 1 / 3 − 1 2 n is 1 3 5 / 3 log 3 ⁡ 8 , the the 5th term from the begging is

Let [x] denote the greatest integer less than or equal to x. If x = ( 3 + 1 ) 5 then [x] is equal to

The coefficient of x 5 in the expansion of ( 1 + 2 x ) 6 ( 1 − x ) 7 is

The sum of the series ∑ r = 0 n ( − 1 ) r n C r 1 2 r + 3 r 2 2 r + 7 r 2 3 r + 15 r 2 4 r + … + m terms is

If ( 5 + 2 6 ) n = I + f ; n , I ∈ N and 0 ≤ f < 1 , then I is equal to

The coefficient of x 4 in the expansion of 1 + x + x 2 + x 3 11 is

Suppose a ∈ R . If the coefficient of x 5 in the expansion of a x + 1 x 3 17 is 680 , then a is equal to

Let a r denote the coefficient of y r – 1 in the expansion of ( 1 + 2 y ) 10 . If a r + 2 a r = 4 , then r is equal to

The remainder when 2 2000 is divided by 17 is

If the 7 th term in the binomial expansion of 3 84 1 / 3 + 3 ln ⁡ x 9 , ( x > 0 ) is equal to 729 , then x is equal to

The coefficient of x 53 in ∑ m = 0 100 100 C m ( x − 5 ) 100 − m ( 4 ) m is

f { p } denotes the fractional part of the number p , then 3 200 8 is equal to

If the sum of the coefficients in the expansion of ( 1 + 2 x ) n is 6561 , then the greatest coefficient in the expansion, is

The coefficients of three consecutive terms of ( 1 + x ) n + 5 are in the ratio 5 : 10 : 14 . Then, n =

The coefficient of x − 10 in ⁡ x 2 − 1 x 3 10 , is

If C r stands for C r n , then the sum of first ( n + 1 ) terms of the ( n + 1 ) terms of the series a C 0 − ( a + d ) C 1 + ( a + 2 d ) C 2 − ( a + 3 d ) C 3 + … , is

If C 0 , C 1 , C 2 , C 3 , … , C n denote the binomial coefficients in the binomial expansion of ( 1 + x ) n then 1. C 1 − 2 ⋅ C 2 + 3 ⋅ C 3 − 4 ⋅ C 4 + … + ( − 1 ) n − 1 n C n =

If the coefficient of r th ( r + 1 ) th and ( r + 2 ) th terms in the expansion of ( 1 + x ) 14 are in A.P ., then the value of r , is

The number of rational terms in the expansion of ( 1 + 2 + 3 3 ) 6 , is

The value of 21 C 1 − 10 C 1 + 21 C 2 − 10 C 2 + 21 C 3 − 10 C 3 + 21 C 4 − 10 C 4 + … + + 21 C 10 − 10 C 10 is

If 1 + x + x 2 + x 3 n = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + … + a 3 n x 3 n , then the value of a 0 + a 4 + a 8 + a 12 + … is

The coefficient of x 7 in the expression 1 + x 10 + x 1 + x 9 + x 2 1 + x 8 + …… + x 10 is-

The coefficient of x 5 in the expansion of ( 1 + x ) 21 + ( 1 + x ) 22 + … + ( 1 + x ) 30 is

If C 0 , C 1 , C 2 , … . . C n denote the coefficients of the binomial expansion ( 1 + x ) n , then the value of C 1 + 3 C 3 + 5 C 5 + … + is

The term independent of x in x 3 + 3 x 2 10 is

In the expansion of ( 1 + x ) 30 , the sum of the coefficients of odd powers of x is

The 9th term of the expansion of 3 x – 1 2 x 8 is

Sum of the last 12 coefficients in the binomial expansion of ( 1 + x ) 23 is

The expression 2 x 2 + 1 + 2 x 2 − 1 6 + 2 2 x 2 + 1 + 2 x 2 − 1 6 is a polynomial of degree

If S n = n C 0 2 + n C 1 2 + n C 3 2 + … + n C n 2 , then maximum value of S n + 1 S n is . (where [.] denotes the greatest integer function)

Let m be the smallest positive integer such that the coefficient of X 2 in the expansion of ( 1 + x ) 2 + ( 1 + x ) 3 + … + ( 1 + x ) 49 + ( 1 + mx ) 50 is ( 3 n + 1 ) 51 C 3 for some positive integer n. Then the value of n is .

The total number of terms in the expansion of ( x + a ) 100 + ( x − a ) 100 after simplification will be

The value of (1.00 2) 12 up to fourth place of decimal is

The ratio of the coefficient of x 10 in ⁡ 1 − x 2 10 and the term independent of x in x − 2 x 10 ,is

The coefficient of x 4 in the expansion of 1 + x + x 2 + x 3 n is

The sum of the coefficients of all even degree terms is x in the expansion of x + x 3 − 1 6 + x − x 3 − 1 6 ( x > 1 ) is equa to

If ( 1 + ax ) n = 1 + 8 x + 24 x 2 + … , t then the values of a and n are

if 1 − x + x 2 n = a 0 + a 1 x + a 2 x 2 + … + a 2 n x 2 n then a 0 + a 2 + a 4 + … + a 2 n is equal to

If the coefficient of second, third and fourth terms in the expansion of (1 + x) 2n are in AP, then 2n 2 – 9n is equal to

If the fourth term in the binomial expansion of x 1 1 + log 10 ⁡ x + x 1 12 6 is equal to 200, and x > 1 then the value of x is

The two successive terms in the expansion of (1 + x) 24 whose coefficients are in the ratio 1:4 are

If the coefficients of x 2 and x 3 are both zero, in the expansion of the expression 1 + ax + bx 2 ( 1 − 3 x ) 15 in powers of x then the ordered pair (a, b) is equal to

If for some positive integer n , the coefficients of three consecutive terms in the binomial expansion of (1 + x) n + 5 are in the ratio 5 : 10 : 14, then the Iargest coefficient in this expansion is

The coefficient of the term independent of x in the expansion of 1 + x + 2 x 3 3 2 x 2 − 1 3 x 9 is

The middle term in the expansion ,of x + 1 2 x 2 n is

The largest term in the expansion of ( 3 + 2 x ) 50 where x = 1 5 is

If the sum of the coefficients in the expansion of (x – 2y + 3 z) n is 128, then the greatest coefficient in the expansion of(1 + x) n is

if n = ( 8 + 3 7 ) 10 , n ∈ N then the least value of n is

The coefficient of x 7 in 1 + 3 x − 2 x 3 10 is equal to

The sum ∑ i = 0 m 10 i 20 m − i , where , p q = 0 , if p < q is maximum, when m is

The coefficients of x 2 y 2 , yzt 2 and xyzt in the expansion of (r + y + z + t) 4 are in the ratio

The last digit of 3 3 4 n + 1 , n ∈ N is

If ( 2 + 3 ) n = I + f where I and n are positive integers and 0 < f < 1 , then ( 1 − f ) ( I + f )

The sum of the series 20 C 0 − 20 C 1 + 20 C 2 − 20 C 3 + … + 20 C 10 is

The coefficient of x 4 in the expansion of 1 + x + x 2 + x 3 6 in powers of x is …….

The coefficient of x 4 in the expansion of 1 + x + x 2 10 is ….

The coefficient of x 17 in the expansion of (x – 1) (x – 2) (x – 3) … (x – 18) is

The fractional part of 2 4 n 15 is

If [x] denotes the greatest integer less than or equal to x, then ( 6 6 + 14 ) 2 n + 1

: Sum of the series S = 3 − 1 10 C 0 − 3 2 10 C 1 + 3 2 10 C 2 − 3 3 10 C 3 + . . . . … + 3 10 10 C 10 is

Let a , b , c , d be any four consecutive coefficients in the binomial expansion ( 1 + x ) n , of then a a + b + c c + d − 2 b b + c is equal to

The coefficient of x 9 in the expansion of E = ( 1 + x ) 9 + ( 1 + x ) 10 + … + ( 1 + x ) 100 is

Suppose a 0 = 2017 , a 1 a 2 , … , a n − 1 , 2023 = a n an are in A . P . Let S = 1 2 n + 1 ∑ r = 0 n n C r a r − 1000 Then S is equal to

If coefficients of r th and ( r + 1 ) th term in the expansion of ( 3 + 2 x ) 74 are equal, then r is equal to:

5 th term from the end in the expansion of x 2 3 − 3 x 2 8 is:

The coefficient of x 53 in the expansion ∑ m = 0 100 100 C m ( x − 3 ) 100 − m 2 m is

For x ∈ R , x ≠ − 1 , if ( 1 + x ) 2016 + x ( 1 + x ) 2015 + x 2 ( 1 + x ) 2014 + … + x 2016 = ∑ i = 0 2016 a j x j then a 17 is equal to

The coefficient of x 4 in the expansion 1 + x + x 2 + x 3 6 in power of x is

The sum of the last eight coefficients in the expansion of ( 1 + x ) 15 is.

If C 0 , C 1 , C 2 , … , C n denote the binomial coefficients i11 the expansion of ( 1 + x ) n then a C 0 + ( a + b ) C 1 + ( a + 2 b ) C 2 + … + ( a + n b ) C n =

The value of ( 2 + 1 ) 5 + ( 2 − 1 ) 5 , is

If for some positive integer n the coefficients of three consecutive terms in the binomial expansion of ( 1 + x ) n + 5 are in the ratio 5 : 10 ; 14 , . then the largest coefficient- in the expansion is

The number of irrational terms in the expansion of ( 5 8 + 2 6 ) 100 , is

2 60 when divided by 7 leaves the remainder

If in the expansion ( 1 + x ) m ( 1 − x ) n , the coefficient of x and x 2 are 3 and -6 respectively then m is

Given positive integer r > 1 , n > 2 and the coefficient of ( 3 r ) th and ( r + 2 ) th terms in the binomial expansion of ( 1 + x ) 2 n are equal. Then

f the coefficient of the 5 t h term be the numerically the greatest coefficient in the expansion of ( 1 − x ) n , then the positive integral value of n , is

The value of ∑ r = 1 15 r 2 15 C r 15 C r − 1 is equal to

The coefficient of x n in the expansion of ( 1 + x ) ( 1 − x ) n , is

7 103 when divided by 25 leaves the remainder

The sum of the series ∑ r = 0 10 20 C r ′ is

The positive integer just greater than ( 1 + 0.0001 ) 10000 , is

Let n be the smallest positive integer such that the coefficients x 2 in the expansion ( 1 + x ) 2 + ( 1 + x ) 3 + … + ( 1 + x ) 49 + ( 1 + m x ) 50 is ( 3 n + 1 ) 51 C 3 for some positive integer n . Then the value of n , is

If C 0 , C 1 , C 2 , C 3 , … , C n are the binomial coe11 ictenis mt e expans of C 0 1 + C 2 3 + C 4 5 + C 6 7 + … has equal to

If C 0 , C 1 , C 2 , … , C n denote the binomial coefficients in the expansion of ( 1 + x ) n then 1 3 ⋅ C 1 + 2 3 ⋅ C 2 + 3 3 ⋅ C 3 + … + n 3 ⋅ C n =

The remainder when 9 103 is divided by 25 is equal to

In the expansion of x 3 − 1 x 2 15 , the constant term, is

The number of terms in the expansion of ( 2 x + 3 y − 4 z ) n , is

If C 0 , C 1 , C 2 , … , C n are binomial coefficients in the expansion of ( 1 + x ) n then the value of C 0 − C 1 2 + C 2 3 − C 3 4 + … + ( − 1 ) n C n n + 1 is

The number of terms in the expansion of 1 + 2 x + x 2 20 when expanded in descending powers of x , is

If C 0 , C 1 , C 2 , … , C n denote the binomial coefficients i11 the expansion of ( 1 + x ) n then a C 0 + ( a + b ) C 1 + ( a + 2 b ) C 2 + … + ( a + n b ) C n =

If A and B are coefficients of x r and x n – r respectively in the expansion of ( 1 + x ) n , then

If C 0 , C 1 , C 2 , … , C n denote the binomial coefficients in the expansion of ( 1 + x ) n then 1 3 ⋅ C 1 + 2 3 ⋅ C 2 + 3 3 ⋅ C 3 + … + n 3 ⋅ C n =

If C 0 , C 1 , C 2 , C 3 , … , C n denote the binomial coefficients in the binomial expansion of ( 1 + x ) n then 1. C 1 − 2 ⋅ C 2 + 3 ⋅ C 3 − 4 ⋅ C 4 + … + ( − 1 ) n − 1 n C n =

If C 0 , C 1 , C 2 , … , C n denote the binomial coefficients in the expansion of ( 1 + x ) n , then C 0 + C 1 2 + C 2 3 + … + C n n + 1 or , ∑ r = 0 n C r r + 1

If the sum of the coefficients of all even powers of x in the product 1 + x + x 2 + ….. + x 2 n 1 − x + x 2 − x 3 + ….. + x 2 n is 61, then n is equal to .

The coefficient of x 4 in the expansion of ( 1 + x + x 2 ) 10 is

If A = a i j 4 × 4 , such that a i j = 2 , when i = j 0 , when i ≠ j , then det ( adj ( adj A ) ) 7 is (where {.} represents fractional part function)

Let 1 + x + 2 x 2 20 = a 0 + a 1 x + a 2 x 2 + … + a 40 x 40 . Then a 1 + a 3 + a 5 + … . + a 37 is equal to

The sum of the co-efficients in the binomial expansion of 1 x + 2 x n is equal to 6561 . Then constant term in the expansion is

T h e n u m b e r o f i n t e g r a l t e r m s i n t h e e x p a n s i o n o f 3 + 5 8 256 i s

T h e c o e f f i c i e n t o f t h e m i d d l e t e r m i n t h e b i n o m i a l e x p a n s i o n i n p o w e r s o f x o f ( 1 + a x ) 4 a n d o f ( 1 – a x ) 6 i s t h e s a m e i f a =

T h e p o s i t i v e i n t e g e r j u s t g r e a t e r t h a n ( 1 + 0 . 0001 ) 1000 i s

I f p a n d q a r e p o s i t i v e r e a l n u m b e r s , t h e n t h e c o e f f i c i e n t s o f x p a n d x q i n t h e e x p a n s i o n o f 1 + x p – q w i l l b e

T h e s u m o f t h e c o e f f i c i e n t s i n t h e e x p a n s i o n o f ( x + y ) n i s 4096 . T h e g r e a t e s t c o e f f i c i e n t i n t h e e x p a n s i o n i s

I f C ( n – 1 , r ) = k 2 – 3 · C ( n , r + 1 ) t h e n k ∈

The sum of coefficients of the expansion 1 x + 2 x n is 6561 . The coefficient of term independent of x is

The last digit, that is, the digit in the units place of the number ( 67 ) 25 – 1 is

If x is so small that its square and higher powers may be neglected, then 1 – x 1 + x 1 2 is approximately equal to

If the second term in the expansion a ‘ 13 + a a – 1 n is 14 a 5 2 then the value of C 3 n C 2 n is

Coefficient of x 3 in the expansion x + 1 x 2 6 is equal to

C 4 20 + 2 C 3 20 + C 2 20 C 18 – 22 is equal to

If x = [ 729 + 6 ( 2 ) ( 243 ) + 15 ( 4 ) ( 81 ) + 20 ( 8 ) ( 27 ) + 15 ( 16 ) ( 9 ) + 6 ( 32 ) 3 + 64 ] [ 1 + 4 ( 4 ) + 6 ( 16 ) + 4 ( 64 ) + 256 ] then x – 1 x is equal to

How many terms are there in the expansion of ( 4 x + 7 y ) 10 + ( 4 x – 7 y ) 10 ?

If n is a natural number, then 4 n – 3 n – 1 is divisible by which one of the following?

If the sum of the coefficients in the expansion of ( x − 2 y + 3 z ) n , n ∈ ℕ is 128 , then the greatest coefficient in the expansion of ( 1 + x ) n is

The smallest integer larger than ( 3 + 2 ) 6 i s

The coefficient of x n in 1 + x 1 ! + x 2 2 ! + ⋯ + x n n ! 2

The remainder when 2 0 + 2 1 + 2 2 + 2 3 + … … + 2 2014 is divided by 9 is

If the coefficient of x 100 in 1 + ( 1 + x ) + ( 1 + x ) 2 + … + ( 1 + x ) n ; ( n ≥ 100 ) is 201 C 101 , then n is equal to

If the coefficients of x 2 and x 3 in expansion of ( 3 + k x ) 9 are equal, then the value of k

If the term free from x is the expansion of x − k x 2 10 is 405 , then the value of k .

The number of terms in the expansion of x 2 + 18 x + 81 15 is

The first three terms in the expansion of 1 + x + x 2 10 are

The coefficient of x 2 y 3 z 4 in ( a x − b y + c z ) 9 is

If the coefficient of x 7 in ax 2 − 1 bx 11 equals the coefficient of x − 7 in ax − 1 bx 2 11 , then a and b satisfy the relation

The term independent of x in the expansion of x 2 − 1 x 6 is

If the coefficients of rth term and ( r + 1 ) th terms in the expansion of ( 1 + x ) 20 are in the ratio 1 : 2 , then r =

If the 5 th term is 24 times the 3 rd term in the expansion of ( 1 + x ) 11 the x =

If the sum of odd terms and the sum of even terms in the expansion of ( x + a ) n are p and q respectively then p 2 + q 2 =

If the coefficients of x 9 , x 10 , x 11 in the expansion of ( 1 + x ) n are in arithmetic progression then n 2 − 41 n =

5 th term of 2 x 2 + 3 x 5 is 10 . Then x =

If ( 7 + 4 3 ) n = I + F where I and n are positive intergers and F is positive proper fraction, then ( I + F ) ( I − F ) =

The numericall6y greatest term in the expansion of ( 3 + 2 x ) 49 when x = 1 / 5 is

C 0 + 4 ⋅ C 1 + 7 . C 2 + . ( n + 1 ) terms =

The greatest coefficient of ( 1 + x ) 10 is

C 0 + C 1 2 + C 2 3 + .. + C n n+1 =

If S n = ∑ i = 0 n 1 n C r and t n = ∑ r = 0 n r n C r then t n S n =

C 0 + C 1 x 2 + C 2 x 2 3 + . . . . . . . . . . . + C n x n n+1 =

The sum of the coefficients of even powers of x in the expansion of 1+x+x 2 +x 3 5 is

The coefficient of x 2 in 1 + x 2 ( 1 − x ) 3 is

If x is so small, higher powers of x may be neglected then x 2 + 27 3 − x 2 + 8 3 =

If x is small so that x 2 and higher powers can be neglected, then the approximate value for ( 1 − 2 x ) − 1 ( 1 − 3 x ) − 2 ( 1 − 4 x ) − 3 is

1 − 1 8 + 1.3 8.16 − 1.3.5 8.16.24 + .. =

Sum of the last 20 coefficients in the expansion of ( 1 + x ) 39 , when expanded in ascending powers of x , is

The remainder when 2 2000 is divided by 17 is

In the expansion of 1 + x + 7 x 11 , the term not containing x is

If the second term of the expansion a 1 / 13 + a a − 1 n is 14 a 5/2 , then the value of n C 3 n C 2 is .

The value of 50 ∑ r = 1 49 2 r 2 − 48 r + 1 ( 50 − r ) ⋅ 50 C r is .

The coefficients of three consecutive terms of (1 + x) n+5 are in the ratio 5 : 10 : 14. Then n = .

Let X = 10 C 1 2 + 2 10 C 2 2 + 3 10 C 3 2 + … + 10 10 C 10 2 where 10 C r , r ∈ { 1 , 2 , … , 10 } denote binomial coefficients. Then the value of 1 1430 X is .

If 1 + x + x 2 48 = a 0 + a 1 x + a 2 x 2 + … + a 96 x 96 then the value of a 0 − a 2 + a 4 − a 6 + … + a 96

The coefficient of x 20 in the expansion of 1 + 1 1 ! x + 1 2 ! x 2 + … + 1 20 ! x 20 3 is

The positive value of λ for which the coefficient of x 2 in the expression x 2 x + λ x 2 10 is 720 is

The coefficient of x 7 in the expression ( 1 + x ) 10 + x ( 1 + x ) 9 + x 2 ( 1 + x ) 8 + … + x 10 is

The sum of the real values of x for which the middle term in the binomial expansion of x 3 3 + 3 x 8 equals 5670 is

lf the term independent of x in the expansion of 3 2 x 2 − 1 3 x 9 is k, then 18k is equal to

The coefficient of x 20 in the expansion of 1 + x 2 40 ⋅ x 2 + 2 + 1 x 2 − 5 is

lf some three consecutive coefficients in the binomial expansion of (x + 1) n in powers of x are in the ratio 2 : 15 : 70, then the average of these three coefficients is

lf a, b and c are the greatest values of 19 C p ′ 20 C q and 21 C r respectively, the

Numerically the greatest term in the expansion of 2 + 3 x 9 when x = 3 2 , is

lntegral part of ( 7 + 4 3 ) n if ( n ∈ N )

For all n€ R, 9 n + 1 – 8n – 9 is divisible by

Series n C 1 + 1 ⋅ n C 2 + 2 ⋅ n C 3 + … + n ⋅ n C n is equal to

The coefficient of x 3 y 4 z 2 in the expansion of ( 2 x − 3 y + 4 z ) 9 is

The coefficient of a 3 b 4 c 5 in the expansion of ( bc + ca + ab ) 6 is

The coefficient of t 24 in the expansion of 1 + t 2 12 1 + t 12 1 + t 24 is

In the polynomial ( x − 1 ) ( x − 2 ) ( x − 3 ) … ( x − 100 ) the coefficient of X 99 is

If in the expansion of ( 1 + x ) m ( 1 − x ) n the coefficient of x and x 2 are 3 and -6 respectively, then m is

The digit at the unit place in the numbers 19 2005 + 11 2005 9 2005 is

if a 1 ,a 2 ,a 3 a, 4 are the coefficients of any four consecutive terms in the expansion of ( 1 + x ) n , then a 1 a 1 + a 2 + a 3 a 3 + a 4 is equal to

if 1 + x − 2 x 2 6 = 1 + a 1 x + a 2 x 2 + … + a 12 x 12 then the expression a 2 + a 4 + a 6 + … + a 12 has the value

If in the expansion of ( a − 2 b ) n ,the sum of 4th and 5th term is zero, then the value of a b is

If the third term in the binomial expansion of 1 + x log 2 ⁡ x 5 equals 2560, then a possible value of x is

If the constant term in the binomial expansion of x − k x 2 10 is 405, then | k | equal

The middle term in the expansion ,of x + 1 2 x 2 n is

The coefficient of x in the expansion of 1 − 3 x + 7 x 2 ( 1 − x ) 16 is

The term independent of x in the expansion of 1 60 − x 8 81 ⋅ 2 x 2 − 3 x 2 6 is equal to

If the middle term of 1 x + xsin ⁡ x 10 is equal to 7 7 8 , then value of x is

If p is a real number and if the middle tern in the expansion of p 2 + 2 8 is 1120,then the value of p is

rf n is even, then the middle term in the expansion of x 2 + 1 x n is 924 x 6 , then n is equal to

The greatest value of the term independent of X, as α varies over R, in the expansion of xcos ⁡ α + sin ⁡ α x 20 i s

The interval in which X must lie, so that the greatest term in the expansion of (1 + x) 2n has the greatest coefficient, is

If a n = ∑ r = 0 n 1 n C r , then ∑ r = 0 n r n C r is equal to

∑ r = 0 n ( − 1 ) r n C r 1 + rx 1 + nx is equal to

30 0 30 10 − 30 1 30 11 + … + 30 20 30 30 is equal to

If ( 1 + x ) n = C 0 + C 1 x + C 2 x 2 + … + C n x n then C 0 2 + C 1 2 + C 2 2 + C 3 2 + … + C n 2 is equal to

the vlaue of 21 C 1 − 10 C 1 + 21 C 2 − 10 C 2 + 21 C 3 − 10 C 3 + 21 C 4 − 10 C 4 + … + 21 C 10 − 10 C 10 is

If n € N, n > 1, then value of E = a − n C 1 ( a − 1 ) + n C 2 ( a − 2 ) + … . . . . + ( − 1 ) n ( a − n ) n C n is

n C 0 − 1 2 n C 1 + 1 3 n C 2 − … + ( − 1 ) n n C n n + 1 is equal to

If ( 1 + x ) n = ∑ r = 0 n C r x r then 1 + C 1 C 0 1 + C 2 C 1 ⋯ 1 + C n C n − 1 is equal to

if C r stand for n C r , the sum of the given series 2 n 2 ! n 2 ! n ! ⋅ C 0 2 − 2 C 1 2 + 3 C 2 2 − … + ( − 1 ) n ( n + 1 ) C n 2 ,where n is an even positive integer, is

The value of − 15 C 1 + 2 ⋅ 15 C 2 − 3 ⋅ 15 C 3 + … . 15 ⋅ 15 C 15 + 14 C 1 + 14 C 3 + 14 C 5 + … + 14 C 11 is

if 1 + x + x 2 n = ∑ r = 0 2 n a r x r , then a 1 − 2 a 2 + 3 a 3 … − 2 na 2 n is equal to

Let R = ( 2 + 3 ) 2 n and f = R − [ R ] where denotes the greatest integer function, then R(L – f ) is equal to

if ( 1 + x ) 15 = C 0 + C 1 x + C 2 x 2 + … + C 15 x 15 then C 2 + 2 C 3 + 3 C 4 + … + 14 C 15 is equal to

if a and d, are two complex numbers, then the sum to (n+ 1) terms of the following series aC 0 − ( a + d ) C 1 + ( a + 2 d ) C 2 − … + … is

If A = 1000 1000 and B = ( 1001 ) 999 then

1 1 ! ( n − 1 ) ! + 1 3 ! ( n − 3 ) ! + 1 5 ( n − 5 ) ! + … is equal to

The coefficient of x n in the polynomial x + n C 0 x + 3 n C 1 x + 5 n C 2 … . . . . x + ( 2 n + 1 ) n C n is

A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of 2 1 3 + 1 2 ( 3 ) 1 3 10 is

The coefficient of x r [ 0 ≤ r ≤ ( n − 1 ) ] in the expansion of ( x + 3 ) n − 1 + ( x + 3 ) n − 2 ( x + 2 ) + ( x + 3 ) n − 3 ( x + 2 ) 2 + … + ( x + 2 ) n − 1 is

The value of x, for which the 6th term in the expansion of 2 log 2 ⁡ 9 x − 1 + 7 + 1 2 ( 1 / 5 ) log 2 ⁡ 3 x − 1 + 1 is 84,is equal to

If n − 1 C r = k 2 − 3 n C r + 1 , then k is belongs to

If sum of the coefficients of the first, second and third terms of the expansion of x 2 + 1 x m is 46, then the coefficient of the term that does not contain x is

The coefficient of x 4 in the expansion of 1 + x + x 2 + x 3 11 is

In the expansion of (1 + x) n , C 1 C 0 + 2 C 2 C 1 + 3 C 3 C 2 + … + n C n C n − 1 is equal to

C 0 + C 1 C 1 + C 2 … C n − 1 + C n is equal to

C 0 C r + C 1 C r + 1 + C 2 C r + 2 + … + C n − r C n is equal to

If the expansion in powers of x of the function 1 ( 1 − ax ) ( 1 − bx ) is a 0 + a 1 x + a 2 x 2 + a 3 x 3 + … then α n is

For natural numbers m, n, if ( 1 − y ) m ( 1 + y ) n = 1 + a 1 y + a 2 y 2 + … and a 1 = a 2 = 10 , then ( m , n ) is

The sum of coefficients of integral powers of r in the binomial expansion of ( 1 − 2 x ) 50 is

If n is a positive integer, then ( 3 + 1 ) 2 n − ( 3 − 1 ) 2 n is

The sum of the coefficients in the expansion of ( 1 + 5 x − 7 x 3 ) 3165 is

If A is the sum of the odd terms and B the sum of even terms in the expansion of (x + a) n , then A 2 − B 2 =

The 7th term in 1 y + y 2 10 ,when expanded in descending power of y, is

The coefficient of x m in ( 1 + x ) m + ( 1 + x ) m + 1 + , … , + ( 1 + x ) n , m ≤ n is

The coefficient of x 5 in the expansion of ( 1 + x 2 ) 5 ( 1 + x ) 4 is

If 7 103 is divided by 25, then the remainder is

The sum of rational terms in the expansion of ( 2 + 3 1 / 5 ) 10 is

If C 0 , C 1 , C 2 , … , C n are the coefficients of the expansion of (1 + x) n , then the value of ∑ 0 n C k k + 1 is

If 7 103 is divided by 25, then the remainder is

The coefficient of x 3 in the expansion of ( 1 − x + x 2 ) 6 is

The two consecutive terms in the expansion of (3x + 2) 74 , whose coefficients are equal, are

If {x} denotes the fractional part of x, then 2 2003 17 is

The sum of the coefficients of all the integral powers of x in the expansion of ( 1 + 2 x ) 80 is

The absolute term (that is, term independent of x ) in the expansion of 3 2 x 2 − 1 3 x 9 is:

Suppose k ∈ R . Let a be the coefficient of the middle term in the expansion of k x + x k 10 , and b be the term independent of x in the expansion of k 2 x + x k 10 . If a b = 1 , then k is equal to

The remainder when 6 n – 5 n is divided by 25 , is

Let 1 + x + x 2 100 = ∑ r = 0 200 a r x r and a = ∑ r = 0 200 a r then value of ∑ r = 1 200 r a r 25 a is-

If the coefficients of three consecutive terms in the expansion of ( 1 + x ) n are in the ratio 1 : 7 : 42 , then value of n is

If a i ( i = 0 , 1 , 2 , . . . 16 ) are real constants such that for every real value of x 1 + x + x 2 8 = a 0 + a 1 x + a 2 x 2 + … + a 16 x 16 then a 5 is equal to

Sum of the last 12 coefficients of in the binomial expansion of ( 1 + x ) 23 is:

If sum of the coefficients in the expansion of ( x + y ) n is 2048 , then the greatest coefficient in the expansion is:

If coefficient of x 3 and x 4 in the expansion of 1 + a x + b x 2 ( 1 − 2 x ) 18 in powers of x are both zeros, then is ( a , b ) equal to

If P ( x ) = 1 3 x + 1 1 + 3 x + 1 5 n − 1 − 3 x + 1 5 n is a 5 t h degree polynomial, then value of n is

The two consecutive terms whose coefficients in the expanssion of ( 3 + 2 x ) 94 are equal, are

The interval in which x ( > 0 ) must lie so that the greatest term in the expansion of ( 1 + x ) 2 n n has the greatest coefficient is

Coefficient of the constant term in the expansion of E = x 2 / 3 + 4 x 1 / 3 + 4 5 1 x 1 / 3 − 1 + 1 x 2 / 3 + x 1 / 3 + 1 − 9 is

Suppose k ∈ R . If the coefficient of x in the expansion of x 2 + k x 5 is 270, then k is equal to:

Suppose p ∈ R . If the fourth term in the expansion of p x + 1 x n is 5 2 then ( n , p ) is equal to:

If the coefficients of the ( r + 2 ) t h and r t h , ( r + 1 ) t h terms in the binomial expansion of ( 1 + y ) m are in , then A . P . m and r satisfy the equation

If the coefficient of ( 3 r + 4 ) t h term and ( 2 r – 2 ) t h term in the expansion of ( 1 + x ) 20 are equal then r is equal to :

Coefficient of x 4 in the expansion of 1 + 3 x + 2 x 2 6 is:

Let R = ( 2 + 1 ) 2 n + 1 , n ∈ N , and f = R − [ R ] , where [ ] denote the greatest integer function, R f is equal to

If sum of the coefficient in the binomial expansion of ( 1 + 2 x ) n is 6561 , then coefficient of the fourth term is :

The coefficient of x 60 in ( 1 + x ) 51 1 − x + x 2 50 is

Coefficient of x 17 in the polynomial P ( x ) = ∏ r = 0 17 ( x + 35 C r ) is

If a = 99 50 + 100 50 and b = 101 50 then

The coefficient of x 20 in the expansion of 1 + 1 1 ! x + 1 2 ! x 2 + … + 1 20 ! x 20 3 is

Let x > − 1 , then statement P ( n ) : ( 1 + x ) n > 1 + n x is true for

If ( 1 + a x ) n = 1 + 10 x + 40 x 2 + … . . , then value of a + n a − n is

If 1 + x − 2 x 2 6 = ∑ r = 0 12 A r x r , then value of A 2 + A 4 + A 6 + … + A 12 is

The sum of the coefficients in the expansion of ( 1 + x − 3 x 2 4321 is

For each n ∈ N , 2 3 n − 1 is divisible by

The greatest integer contained in ( 3 + 1 ) 6 is

The ratio of the coefficient of x 10 in 1 + x 2 10 and the term independent of x in x + 2 x 10 is

The coefficient of x 7 in the expansion of 1 + 1 1 ! x + 1 2 ! x 2 + 1 3 ! x 3 + 1 4 ! x 4 + 1 5 ! x 5 2 is

The number of terms in the expansion of x 2 + 6 x + 9 ) 30 is

If coefficient of 9th, 10th and 11th terms of ( 1 + x ) n are in A . P . , then value of n can be

For each n ∈ N , x 2 n − 1 + y 2 n − 1 is divisible by

The expansion of ( x + y + z ) n is given by

( p + 2 ) th term from the end in the binomial expansion of x 2 − 2 x 2 2 n + 1 is

Sum of the series ∑ k = 1 ∞ ∑ r = 0 k 2 2 r 7 k k C r is

The number of terms in the expansion of the multinomial ( x + y + z ) n

If a n = 7 + 7 + 7 + ⋯ having n radical signs then by methods of mathematical induction which of the following is true

Let C r = 15 C r , 0 ≤ r ≤ 15 . Sum of the series S = ∑ r = 1 15 r C r C r − 1 is

The number of terms in the expansion of ( x 1 + x 2 + . . . . . . + x r ) n is

If the 6th term from the beginning is equal to the 6th term from the end in the expansion of 2 1 / 5 + 1 3 1 / 5 n , then n is equal to

If the coefficients of x 7 and x 8 in 2 + x 3 n are equal, then value of n is

If the middle term of x 2 + 1 x n is 924 x 6 , then value of n is

If the coefficient of x 7 in the expansion of a x 2 + 1 b x 11 and the coefficient of x – 7 is the expansion of a x − 1 b x 2 11 are equal, then

In the expansion of x 3 − 1 x 2 15 , the constant term equals

The greatest term in the expansion of 5 1 + 1 5 20 is :

If x + y = 1 , then value of ∑ r = 0 n ( r ) n C r x n − r y r is

If a n = 2 2 n + 1 then for n > 1 , last digit of a n is

Let S ( θ ) denote the sum of coefficients in the expansion of 2 − x sin ⁡ θ + x 2 cos ⁡ θ 2 n . Maximum value of S ( θ ) is

If sum of the coefficients in the expansion of x + 1 x n is 128., then coefficient of x in the expansion of x + 1 x n is

If a is real and the 4th term in the expansion of a x + 1 x n is 5 / 2 , for each x ∈ R – { 0 } , then values of n and a are respectively

Suppose ( x + a ) n = T 0 + T 1 + T 2 + … + T n Then T 0 − T 2 + T 4 − … 2 + T 1 − T 3 + T 5 − … 2 is equal to

If the coefficients of 5 th , 6 th and 7 th terms in the expansion of ( 1 + x ) n are in A.P., then n =

If, in the binomial expansion of ( a − b ) n , n ≥ 5 he sum of 5th and 6th term is zero, then value of a b is

If A and B are coefficients of x n in the expansions of ( 1 + x ) 2 n and ( 1 + x ) 2 n − 1 respectively, then

If the ratio of the 7th term from the beginning to the 7th term from the end in the expansion of 2 1 / 3 + 3 − 1 / 3 n is 1 / 6 then the value of n is

The coefficient of x 5 in the expansion of ( 1 + x ) 21 + ( 1 + x ) 22 + ( 1 + x ) 23 + . . . . . . . . . . . + ( 1 + x ) 30 is

If the ( r + 1 ) t h term in the expansion of a 1 / 3 b 1 / 6 + b 1 / 2 a 1 / 6 21 has equal exponents of both a and b , then value of r is

If C r = n C r , then the value of 1 + C 1 C 0 1 + C 2 C 1 ⋯ 1 + C n C n − 1

The last term in the binomial expansion of 2 − 1 2 n is 1 ( 3 ) 9 1 3 log ⁡ 3 then the 5th term from the beginning is

If the number of terms in the expansion of 1 + 5 x + 10 x 2 + 10 x 3 + 5 x 4 + x 5 20 is m , then unit’s place of 2 m is

If sum of the coefficients in the expansion 2 x 2 − 3 c x + c 2 ) 17 is zero, then c is equal to

For 0 ≤ r < 2 n , 2 n + r C n 2 n − r C n cannot exceed

If C r = n C r , then the value of 1 + C 1 C 0 1 + C 2 C 1 ⋯ 1 + C n C n − 1

Coefficient of the term independent of x in the expansion of x + 1 x 2 / 3 − x 1 / 3 + 1 − x − 1 x − x 1 / 2 10 is

Coefficient of constant term in the expansion of x 3 + 5 ( 3 ) − log ⁡ 3 x 3 4 is

The value of 2 C 0 + 2 2 2 C 1 + 2 3 3 C 2 + … + 2 11 11 C 10 , is

In the expansion of x 3 − 1 x 2 n , n ∈ N , if the sum of the coefficients of x 5 and x 10 is 0, then n is

Let ( 1 + x ) n = C 0 + C 1 x + C 2 x 2 + … + C n x n and C 1 C 0 + 2 C 2 C 1 + 3 C 3 C 2 + ⋯ + n C n C n − 1 = 1 k n ( n + 1 ) , then value of k is

The coefficient of x 28 in the expansion of 1 + x 3 − x 6 3 n is

If C r = n C r , then C 0 − 1 3 C 1 + 1 5 C 2 ⋯ upto ( n + 1 ) term equal

Coefficient of t 24 in the expansion of 1 + t 2 12 1 + t 12 1 + t 24 is

Degree of the polynomial x 2 − 1 − x 3 4 + x 2 + 1 − x 3 4

The remainder when 7 128 is divided by 10 is

The value of ∑ k = 0 n 4 n + 1 C k + 4 n + 1 C 2 n − k is

The coefficient of x 5 in the expansion of ( 1 + x ) 21 + ( 1 + x ) 22 + ⋯ + ( 1 + x ) 30 is

If ( 1 + x ) 1 + x + x 2 1 + x + x 2 + x 3 … ( 1 + x + x 2 + … + x n = a 0 + a 1 x + a 2 x 2 + … + a m x m then the value of a 1 is

Coefficient of the term independent of x in the expan-sion of ( 1 + x ) 2 n x 1 − x − 2 n is :

Coefficient of 1 x in the expansion of ( 1 + x ) n ( 1 + 1 / x ) n is

Let 1 + x 2 2 ( 1 + x ) n = ∑ r = 1 n + 4 a r x r . If a 1 , a 2 , a 3 are in A.P., then n is equal to

The coefficient of a 8 b 4 c 9 d 9 in ( abc + abd + acd + bcd ) 10 is

If x 2 k occurs in the expansion of x + 1 x 2 n − 3 , then

If the value of the third term in the expansion of x + x log 10 ⁡ x 5 is 10 6 , then x may have value(s)

If ( 1 + x ) n = C 0 + C 1 x + C 2 x 2 + … + C n x n , the value of C 0 + 2 C 1 + 3 C 2 + … + ( n + 1 ) C n is

The greatest term in the expansion of ( 1 + x ) 10 when x = 2 / 3 is

The coefficient of x 10 in the expansion of 1 + x 2 − x 3 8 is

If in the expansion of ( 1 + x ) n , a , b , c are three consecutive coefficients, then n =

The greatest term in the expansion of ( 3 + 5 x ) 15 , when x = 1 / 5 is

Let P n denote the product of all the coefficients in the expansion of expansion of ( 1 + x ) n . If ( 20 ) ! P n + 1 = 21 20 P n , then n is equal to

The coefficient of x 2 y 3 in the expansion of ( 1 − x + y ) 20 is

Coefficient of x 4 in 1 + x + x 2 + x 3 11 is-

The sixth term in the expansion of x 1 / 3 + y 1 / 2 n if the binomial coefficient of the third term form the end is 45, is

The value of 19 3 + 6 3 + ( 3 ) ( 19 ) ( 6 ) ( 25 ) 3 6 + 6 ( 243 ) ( 2 ) + ( 15 ) ( 81 ) ( 4 ) + ( 20 ) ( 27 ) ( 8 ) + 15 ( 9 ) ( 16 ) + 6 ( 3 ) ( 32 ) + 64 is

Coefficient of x 4 in the expansion of x 2 − 3 x 2 10 is

In the expansion of ( x + a ) n , if the sum of odd terms be A and the sum of even terms be B, then value of ( x 2 – a 2 ) n is

If sum of the coefficients of x 7 and x 4 in the expansion of x 2 a − b x 11 is zero, then

If n ∈ N , n > 1 , then value of E = a − n C 1 ( a − 1 ) + n C 2 ( a − 2 ) + … + ( − 1 ) n ( a − n ) n C n is

1.2.3 + 2.3.4 + 3.4.5 + … upto n terms is equal to

The positive integer just greater than S = ( 1 + 0 . 0001 ) 10000 is

The total number of terms in the expansion of ( x + a ) 200 + ( x − a ) 200 after simplification is

If n > 2 , r > 1 and the coefficients of ( 3 r ) th and ( r + 2 ) th terms in the expansion of ( 1 + x ) 2 n are equal, then

If the coefficients of ( 2 r + 4 ) th and ( r – 2 ) th terms in the expansion of ( 1 + x ) 18 are equal, then value of r is

x n = a 0 + a 1 ( 1 + x ) + a 2 ( 1 + x ) 2 + … + a n ( 1 + x ) n = b 0 + b 1 ( 1 − x ) + b 2 ( 1 − x ) 2 + … + b n ( 1 − x ) n then for n = 101 , a 50 , b 50 equals:

Coefficient of the term independent of x in the expansion x + 1 x 4 x − 1 x 12 is

If the third term in the expansion of 1 x + x log 10 ⁡ x 5 x > 1 , is 1000 , then x equals

If a > 0 and coefficient of x 2 , x 3 , x 4 in the expansion of 1 + x a 6 are in A . P . , then a equals

Suppose ABC is a triangle and n is a natural number, then sum of the series S = ∑ r = 0 n n C r a n − r b r cos ⁡ [ n A − ( n − r ) B ] is

If α and β are the coefficients of x r and x n – r respec-tively in the expansion of ( 1 + x ) n , then

If ( 1 + p x ) n = 1 + 24 x + 2642 x 2 + … . . . then

If a i ( i = 0 , 1 , 2 , … , 16 ) be real constants such that for every real value of x , 1 + x + x 2 8 = a 0 + a 1 x + a 2 x 2 + … + a 16 x 16 , then a 5 is equal to:

The coefficient of t 24 in 1 + t 2 12 1 + t 12 1 + t 24 is

The coefficient of the middle term in the binomial expansion of ( 1 + a x ) 4 and of ( 1 – a x ) 6 6 is the same is a equals

If the sum of the coefficients in the expansion of ( x + y ) n is 2048, then the greatest coefficient in the expansion is :

The coefficient of x 5 in the expansion of 1 + x 2 5 ( 1 + x ) 4 is

For natural numbers m and n ( 1 − y ) m ( 1 + y ) n = 1 + a 1 y + a 2 y 2 + … and a 1 = a 2 = 10 , then ( m , n ) equals

Sum of two middle terms in the expansion of ( 1 + x ) 2 n − 1 is

Sum of the series ∑ r = 0 10 ( − 1 ) r 10 C r 1 2 r + 3 r 2 2 r is

Sum of the last 30 coefficients of powers of x in the binomial expansion of ( 1 + x ) 59 is

The expansion of x + x 3 − 1 5 + x − x 3 − 1 5 is a polynomial of degree

If a is positive real number and if the middle term of a 3 + 3 8 is 1120 then value of a is

The expression x + x 3 − 1 6 + x − x 3 − 1 6 is a polynomial of degree

The remainder when 2 2000 is divided by 17 is

If in the binomial expansion of ( 1 − x ) m ( 1 + x ) n , the coefficients of x and x 2 are respectively 3 and -4, then the ratio m : n is equal to

The sum of the series S = 20 C 0 − 20 C 1 + 20 C 2 − 20 C 3 + … + 20 C 10 is

The coefficient of x r in the expansion of S = ( x + 3 ) n − 1 + ( x + 3 ) n − 2 ( x + 2 ) + ( x + 3 ) n − 3 ( x + 2 ) 2 + … + ( x + 2 ) n − 1 is

If m and n are positive integers, then value of m C 0 n C k + m C 1 n C k − 1 + … + m C k n C 0

For a positive integer n if the mean of the binomial coefficients in the expansion of ( a + b ) 2 n − 3 is 16, then n is equal to :

The number of distinct terms in the expansion of ( 1 + 3 x + 3 x 2 + x 3 7 is

Number of zeros at then end of 99 1001 + 1 is

The remainder when 8 2 n − 62 2 n + 1 is divided by 9 is

The coefficient of x k in the expansion of E = 1 + ( 1 + x ) + ( 1 + x ) 2 + … + ( 1 + x ) n is

If x > 0 , then the number of positive terms in the expansion of ( 1 + i x ) 4 n is

The number of terms whose values depend on x in the expansion of x 2 − 2 + 1 x 2 n is

The coefficient of x 7 in the expansion of 1 − x − x 2 + x 3 6 is

If the coefficients x 7 in ax 2 + 1 bx 11 and coefficient of x -7 in ax − 1 bx 2 11 are equal, then the value of ab is .

If the coefficients of the r th , ( r + 1 ) th , ( r − 2 ) th terms in the expansion of ( 1 + x ) 14 are in AP, then the the sum of all possible values of r is .

Degree of the polynomial x 2 + 1 + x 2 − 1 8 + 2 x 2 + 1 + x 2 − 1 8 is .

If the middle term in the expansion of x 2 + 2 8 is 1120, then the product of possible real values of x is .

The coefficient of x 53 in ∑ m = 0 100 100 C m ( x − 5 ) 100 − m ( 4 ) m is

The coefficient of x n i n t h e e x p a n s i o n o f 1 + 1 1 ! x + 1 2 ! x 2 … + 1 n ! x n 2

In the expansion of ( 1 + p x ) n , n ∈ N , the coefficient of x and x 2 are 8 and 24 respectively, then

The coefficient of x n i n t h e e x p a n s i o n o f 1 + 1 1 ! x + 1 2 ! x 2 … + 1 n ! x n 2

The greatest value of the term independent of x , as a varies over R , in the expansion of x cos ⁡ α + sin ⁡ α x 20 is

Let E = 1 2017 + 2 2017 + 3 2017 + … + 2016 2017 then E is divisible by

Sum of the last 20 coefficients in the expansion of ( 1 + x ) 39 , when expanded in ascending power of x is

The coefficient of x n in the expansion of ( 1 + x ) ( 1 − x ) n

If ( 1 + x ) 15 = C 0 + C 1 x + C 2 x 2 + … + C 15 x 15 then value of the expression S = C 2 + 2 C 3 + 3 C 4 + … + 14 C 15 is

If a i ( i = 0 , 1 , 2 , … 16 ) are real constants such that for every real value of x 1 + x + x 2 8 = a 0 + a 1 x + a 2 x 2 + … + a 16 x 16 , then a 5 is equal to

Sum of the last 12 coefficients of in the binomial expansion of ( 1 + x ) 23 is:

If coefficient of x 3 and x 4 in the expansion of 1 + a x + b x 2 ( 1 − 2 x ) 18 in powers of x are both zeros, then ( a , b ) is equal to

If denotes the greatest integer function, then ( 2 + 1 ) 6 is equal

Let a, b, c, d be any four consecutive coefficients in the binomial expansion of ( 1 + x ) n ,then a a + b + c c + d − 2 b b + c is equal to

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 x 20 is

If the coefficients of the r t h , ( r + 1 ) t h and ( r + 2 ) t h terms in the binomial expansion of ( 1 + y ) m are in A.P., then m and r satisfy the equation

If x n = a 0 + a 1 ( 1 + x ) + a 2 ( 1 + x ) 2 + … …….. + a n ( 1 + x ) n = b 0 + b 1 ( 1 − x ) + b 2 ( 1 − x ) 2 + … + b n ( 1 − x ) n then for n = 201 , a 101 , b 101 is equal to:

For each n ∈ N , 2 3 n − 1 is divisible by

The interval in which x ( > 0 ) must lie so that the greatest term in the expansion of ( 1 + x ) 2 n has the greatest coefficient is

For all n ∈ N , n ( n + 1 ) ( 2 n + 1 ) is divisible by

For each n ∈ N , x 2 n − 1 + y 2 n − 1 is divisible by

If a n = 2 2 n + 1 , then for n > 1 last digit of a n is

Degree of the polynomial x 2 − 1 − x 3 4 + x 2 + 1 − x 3 4 , is

The number of rational terms in the expansion of ( 1 + 2 + 5 3 ) 6 is

Sum of two middle terms in the expansion of ( 1 + x ) 2 n − 1 is

If a is a real number and if the middle term of a 3 + 3 8 is 1120, then value of a is

If coefficient of a 2 b 3 c 4 in ( a + b + c ) m ( where m ∈ N ) is L ( L ≠ 0 ) , then in same expansion coefficient of a 4 b 4 c 1 will be

The number of values in set of values of r for which 23 C r + 2 ⋅ 23 C r + 1 + 23 C r + 2 ≥ 25 C 15 is .

If the constant term in the binomial expansion of x 2 − 1 x n , n ∈ N is 15, then the value of n is equal to .

The last two digits of the number 3 400 are

The value of 15 C 0 2 – 15 C 1 2 + 15 C 2 2 − ⋯ − 15 C 15 2 is

The coefficient of x 5 in the expansion of 1 + x 2 5 ( 1 + x ) 4 is

If the coefficients of r t h and ( r + 1 ) t h terms in the expansion of ( 3 + 7 x ) 29 are equal, then r equals

The expression 1 4 x + 1 1 + 4 x + 1 2 7 − 1 − 4 x + 1 2 7 is a polynomial in x of degree.

The number of integral terms in the expansion of ( 5 1 / 2 + 7 1 / 8 1024 , is

The number of terms which are free from radical signs in the expansion of y 1 / 5 + x 1 / 10 55 , is

In the binomial expansion of ( 1 + a ) m + n n , if the coefficients of a m and a n are A and B respectively, then

If 1 + 2 x + 3 x 2 10 = a 0 + a 1 x + a 2 x 2 + … + a 20 x 20 , then a 1 equals

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