Let 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 such that . If the probability of the event E is where , then is equal to [2026]

Question

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 __________ . [2026]

Smart Achievers · Accepted Answer

(15)

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

$P (S) contains 2^{5} = 32 elements$

$both set A and set B are subsets of P (S)$

$Every element has 4 choices$

$Favourable cases = 3^{5}$

$Total cases = 4^{5}$

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

$m = 5, n = 10$

$m + n = 15$

Solution

Ans.
(15)

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

$P (S) contains 2^{5} = 32 elements$

$both set A and set B are subsets of P (S)$

$Every element has 4 choices$

$Favourable cases = 3^{5}$

$Total cases = 4^{5}$

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

$m = 5, n = 10$

$m + n = 15$

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Solution

Q. 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∩B=∅. If the probability of the event E is 3p2q, where p,q∈ℕ, then p+q is equal to __________ . [2026]

Ans. (15) S={a,b,c,d,e} P(S) contains 25=32 elements both set A and set B are subsets of P(S) Every element has 4 choices Favourable cases=35 Total cases=45 P=3545=35210 m=5, n=10 m+n=15

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Ans.
(15)

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

$P (S) contains 2^{5} = 32 elements$

$both set A and set B are subsets of P (S)$

$Every element has 4 choices$

$Favourable cases = 3^{5}$

$Total cases = 4^{5}$

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

$m = 5, n = 10$

$m + n = 15$