Two cards are drawn at random from a well-shu | vedclass

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(B) Total number of ways to draw $2$ cards from $52$ is $^{52}C_2 = \frac{52 	imes 51}{2} = 1326$. Prime numbers in a suit are $2, 3, 5, 7$ (total $4

Vedclass · Accepted Answer

$\frac{62}{663}$

Vedclass · Answer

(B) Total number of ways to draw $2$ cards from $52$ is $^{52}C_2 = \frac{52 	imes 51}{2} = 1326$. Prime numbers in a suit are $2, 3, 5, 7$ (total $4$ cards per suit, $16$ in total). Multiples of $5$ in a suit are $5$ and $10$ (total $2$ cards per suit, $8$ in total). Note that $5$ is both a prime number and a multiple of $5$. Let $A$ be the set of prime cards ($16$ cards) and $B$ be the set of multiples of $5$ ($8$ cards). The intersection $A \cap B$ contains the four $5$s. We want one card from $A$ and one from $B$. Case $1$: One card is $5$ and the other is a prime other than $5$. Number of ways = $^4C_1 	imes ^{12}C_1 = 4 	imes 12 = 48$. Case $2$: One card is $5$ and the other is a multiple of $5$ other than $5$ (i.e.,$10$). Number of ways = $^4C_1 	imes ^4C_1 = 4 	imes 4 = 16$. Case $3$: One card is a prime other than $5$ and the other is a multiple of $5$ other than $5$. Number of ways = $^{12}C_1 	imes ^4C_1 = 12 	imes 4 = 48$. Total favorable outcomes = $48 + 16 + 48 = 112$. Wait, re-evaluating: The selection is $1$ prime and $1$ multiple of $5$. Favorable outcomes = $(16 	imes 8) - (	ext{overlap}) = 128 - 4 = 124$. Probability = $\frac{124}{1326} = \frac{62}{663}$. Thus, option $(b)$ is correct.

Two cards are drawn at random from a well-shuffled pack of $52$ cards. Find the probability that one card is a prime number and the other is a multiple of $5$.

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