Question

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

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

Answer 1

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

$S=\{1,2,3,...,100\}$
total number of non-empty subsets = $2^{100}-1$
toal subsets having odd product = $2^{50}-1$
$\therefore$ total subset having even product
$=(2^{100}-1)-(2^{50}-1)$
$=2^{100}-2^{50}$
$=2^{50}(2^{50}-1)$