JEE Main 2026 — Jan 24, Shift 2 — Mathematics solution

JEE Main2026Jan 24, Shift 2Mathematics

Let S be a set of 5 elements and $P (S)$ denote the power set of S. Let E be an event of choosing an ordered pair $(A, B)$ from the set $P (S) \times P (S)$ such that $A \cap B = \emptyset$ . If the probability of the event E is $\frac{3 ^{p}}{2 ^{q}}$ , where $p, q \in N$ , then $p + q$ is equal to ________.

Your answer

Solution

For each element of $S$ , there are 4 possibilities regarding subsets $(A, B)$ : in both, in $A$ only, in $B$ only, in neither. Total ordered pairs $= 4^{5} = 2^{10}$ .

For $A \cap B = \emptyset$ , the "in both" option is excluded, leaving 3 choices per element. Favourable pairs $= 3^{5}$ .

$P (E) = \frac{3 ^{5}}{4 ^{5}} = \frac{3 ^{5}}{2 ^{10}}$ , so $p = 5$ and $q = 10$ .

$p + q = 15$ .

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