Which of the following statement(s)is/are CORRECT?



 

a) A flight of stairs has 10 steps. A person can go up the steps one at a time, two at a time or any combination of 1s and 2s. Let x + 2y = 10
where x is the number of times he takes single steps and y is the number of times he takes two steps.

Hence, total no. of ways

= 1+ 15 + 35 + 28 + 9 + 1 = 89.

a) KEY : C
If the number is 100a + 10b + c where $1\le a\le 9$
$0\le b,c\le 9$

$\text{ then }a=\frac{b+c}{2}\text{ or }b=\frac{a+c}{2}\text{ or }c=\frac{a+b}{2}\text{, }$

where a, b, c are distinct 

c) Let m be a positive integer for which

$n^2+96=m^2$

$(m+n)(m-n)=96\Rightarrow (m+n)(m+n-2n)=96$

$(m+n)(m-n)\text{ must be both even as }$

$96=2\times 48\text{ or }4\times 24\text{ or }6\times 16\text{ or }8\times 12$

d) No. is of type 2014.1, 2014.12, 2014.123, etc. So the question involves selection of 1 or more numbers
out of 1,2,.....9 and arranging them in increasing order hence answer $=2^9-1.$