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 $\frac{3^{\mathrm{p}}}{2^{\mathrm{q}}}$, where $\mathrm{p}, \mathrm{q} \in \mathbb{N}$, then $\mathrm{p}+\mathrm{q}$ is equal to ______.

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Probability

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 ______.

Answer: 15

The power set

P (S)

has

2^{5} = 32

elements. For an ordered pair

(A, B)

with

A \cap B = \emptyset

, each of the

5

elements independently lies in

A

only, in

B

only, or in neither, giving

3^{5}

favourable pairs out of

(2^{5})^{2} = 2^{10}

total. Hence

P (E) = \frac{3 ^{5}}{2 ^{10}} = \frac{3 ^{p}}{2 ^{q}}

with

p = 5

and

q = 10

. Therefore

p + q = 15

JEE Main · 2026 · 24 Jan 2026 · Shift 2 · Integer answer