JEE Mathematics 2026 Question with Solution

Given: $S$ is the set of the first $11$ natural numbers.

Find: The number of subsets $B \subseteq S$ such that $n(B) \ge 2$ and the product of all elements of $B$ is even.

A product is even if the subset contains at least one even number.

The set $S$ contains $5$ even numbers,

and $6$ odd numbers.

Total number of subsets of $S$ is

Subsets containing only odd numbers are formed from the $6$ odd elements, so their number is

Therefore, subsets having at least one even element are

Now remove subsets with fewer than $2$ elements. The empty set is already excluded because it does not contain an even element. The only invalid subsets here are the single-element even subsets, and their number is $5$.

Hence, the required number of subsets is

Therefore, the required number of elements is $1979$.

Given: We need subsets of $S$ with size at least $2$ and even product.

Find: The required count.

Instead of counting valid subsets directly, count all subsets with at least one even element, then exclude the single-element even subsets.

Since a subset has even product exactly when it contains at least one even number,

The shortcut works because the complement of "contains an even number" is "contains only odd numbers," which is much easier to count.