JEE Main 2026 — Jan 22, Shift 2 — Mathematics solution

JEE Main2026Jan 22, Shift 2Mathematics

Let $S$ be the set of the first 11 natural numbers. Then the number of elements in $A = {B \subseteq S : n (B) \geq 2$ and the product of all elements of $B$ is even $}$ is ______.

Your answer

Solution

$S = {1, 2, \dots, 11}$ contains 6 odd and 5 even numbers. Total subsets: $2^{11} = 2048$ .

Subsets with odd product = subsets of the 6 odd numbers $= 2^{6} = 64$ . These all have even-product-violating structure.

Subsets with even product (any size): $2048 - 64 = 1984$ .

We require $n (B) \geq 2$ , so exclude singletons with even product. There are 5 such singletons (one for each even number).

Answer: $1984 - 5 = 1979$ .

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