JEE Main 2026 — Sets and Relations Question with Solution

The set $S=\{1,2,3,...,11\}$ has 11 elements.
We need to count subsets $B$ with $n(B)\ge 2$ and even product.
A product is even iff at least one element is even.
Equivalently, count complements: a product is odd iff all elements are odd.
The odd numbers in $S$ are $\{1,3,5,7,9,11\}$ (6 elements).
Subsets of odd numbers: $2^6=64$.
Among these, those with $n(B)\ge 2$ are all except empty set (1) and singletons (6): $64-7=57$.
Total subsets with $n(B)\ge 2$: $2^{11}-1-11=2036$.
Subsets with even product and $n(B)\ge 2$: $2036-57=1979$.