Question

Let $S=\{1,2,3,\dots ,100\}.$ The number of non-empty subsets $A$ of $S$ such that the product of elements in $A$ is even is :-

• A. $2^{50}(2^{50}-1)$
• B. $2^{100}-1$
• C. $2^{50}-1$
• D. $2^{50}+1$

Answer 1

The correct option is A $2^{50}(2^{50}-1)$

$S=\{1,2,3-\dots -100\}$
= Total non empty subsets-subsets with product of element is odd
$=2^{100}-1-1\left[(2^{50}-1)\right]$
$=2^{100}-2^{50}$
$=2^{50}(2^{50}-1)$