Question

If A and B be two finite sets such that the total number of subsets of A is $960$ more than the total number of subsets of B, then $n\left(A\right)-n\left(B\right)$ (where n(x) denotes the number of elements in set x) is equal to?

Answer 1

let the number of elements in A be m and B be n

therefore the total number of subsets of A is $2^m$ and number of subsets of B is $2^n$

given $2^m-2^n=960$

we know from this equation that m>n

therefore taking n common we get

$2^n(2^{m-n}-1)=960$

as $2^{(}m-n)-1$ is odd the even part is only $2^n$

960 can be written as $2^6\times 15$

therefore from above equation we can observe that $n=5$

and $2^{(}m-5)-1$=15$

$\Rightarrow 2^{(}m-5)=16$ so m=9

therefore n(A)-n(B)=m-n=4