Question

Let $S = \{1,2,3,..., 100\}$. If the number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is $2^x (2^y - 1)$, then $x + y =$

Answer 1

$\because$ Product of two even number is always even and product of two odd numbers is always odd.
$\therefore $ Number of required subsets = Total number of subsets - Total number of subsets having only odd numbers
$ = 2^{100} - 2^{50}$
$= 2^{50}(2^{50}- 1)$