JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Permutations and Combinations

The product of the elements of $B$ is even precisely when $B$ contains at least one even number. Among $S=\{1,2,\dots ,11\}$ there are $5$ even numbers and $6$ odd numbers. The number of subsets of $S$ containing at least one even number is $2^{11}-2^6=2048-64=1984$, since the $2^6$ subsets of the odd elements are exactly those with no even number. From these remove the subsets with fewer than two elements: the empty set already contains no even number, while the single-element subsets with an even element number $5$. Hence the number of subsets with $n(B)\ge 2$ and even product is $1984-5=1979$.