Sets, Relations and Functions – Permutations and Combinations – JEE Main 22 Jan 2026 Shift 2

Solution:

Let $S = \{1, 2, 3, \dots, 11\}$. The total number of elements $n(S) = 11$.
The set $A$ contains subsets $B$ of $S$ such that:
1. The number of elements in $B$ is at least 2 ($n(B) \ge 2$).
2. The product of elements in $B$ is even.

The condition “Product is even” is equivalent to “At least one element in B is even”.
It is easier to calculate the total valid subsets and subtract those where the product is ODD.
Product is odd $\Leftrightarrow$ All elements in $B$ are odd.

**Calculate Total Subsets with size $\ge 2$**
Total subsets of $S = 2^{11} = 2048$.
Subsets with size 0 (Empty set) = $\binom{11}{0} = 1$.
Subsets with size 1 = $\binom{11}{1} = 11$.
Total subsets with $n(B) \ge 2$:
$$ N_{total} = 2048 – 1 – 11 = 2036 $$

**Calculate Subsets with odd product and size $\ge 2$**
Odd numbers in $S$: $\{1, 3, 5, 7, 9, 11\}$. There are 6 odd numbers.
Total subsets formed using only odd numbers = $2^6 = 64$.
Among these, we must exclude those with size $< 2$: Size 0 (Empty set) = $\binom{6}{0} = 1$. Size 1 (Single odd numbers) = $\binom{6}{1} = 6$. Subsets with only odd numbers and $n(B) \ge 2$: $$ N_{odd} = 64 - 1 - 6 = 57 $$
**Calculate Required Subsets**
We subtract the purely odd subsets from the total valid subsets:
$$ N_{even\_product} = N_{total} – N_{odd} $$
$$ N_{even\_product} = 2036 – 57 = 1979 $$

Ans. (1979)