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 ) × P ( S ) such that A cap B =varnothing . If the probability of the event E is frac 3 p 2 q , where p, q in mathbb N , then p+q is equal to

Mathematics · Sets and Relations

JEE Main 2026 — 24 January, Evening Shift — Question 24

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

Answer: 15

Numerical answer — enter this value.

Step-by-step solution

$\mathrm{S}=\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}, \mathrm{e}\}$

contains 32 elements both set A and set B are subsets of $\mathrm{P}(\mathrm{S})$

Every element has 4 choices .

Favourable cases $=3^{5}$

Total cases $=4^{5}$

$\mathrm{P}=\frac{3^{5}}{4^{5}}=\frac{3^{5}}{2^{10}}$

$\mathrm{p}=5, \mathrm{q}=10$

$\mathrm{p}+\mathrm{q}=15$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 24 January, Evening Shift
Subject: Mathematics
Chapter: Sets and Relations
Topic: Sets