Let S = { 1, 2, 3, .......... 100 }. The number of nonempty subsets A

$S=\{1,2,3 \cdots-100\}$

$=$ Total non empty subsets-subsets with product of element is odd

$=2^{100}-1-1\left[\left(2^{50}-1\right)\right]$

$=2^{100}-2^{50}$

$=2^{50}\left(2^{50}-1\right)$