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

Answer: 100

$\because$ 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
$=2^{100}-2^{50}$
$=2^{50}(2^{50}-1)$