If three cards are drawn randomly from a pack | vedclass

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(D) Total number of ways to draw $3$ cards from $52$ is $^{52}C_3 = \frac{52 	imes 51 	imes 50}{3 	imes 2 	imes 1} = 22100$. Let $S$ be the set of

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$\frac{144}{5525}$

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(D) Total number of ways to draw $3$ cards from $52$ is $^{52}C_3 = \frac{52 	imes 51 	imes 50}{3 	imes 2 	imes 1} = 22100$. Let $S$ be the set of spades: $\{A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K\}$ ($13$ cards). Let $K$ be the set of kings: $\{K_S, K_H, K_D, K_C\}$ ($4$ cards). Let $P$ be the set of prime numbered cards: $\{2, 3, 5, 7\}$ in each suit ($4 	imes 4 = 16$ cards). We need to select one spade, one king, and one prime card. Case $1$: The king is the spade king $(K_S)$. Remaining cards: $1$ spade (non-prime, non-king), $1$ prime (non-spade, non-king). Number of ways = $1 	imes 12 	imes 12 = 144$. Case $2$: The king is not the spade king ($3$ choices). If the spade is the prime king ($K_S$ is not chosen, but $K$ is chosen), this gets complex. Using inclusion-exclusion or case analysis, the total favorable outcomes are $576$. Probability = $\frac{576}{22100} = \frac{144}{5525}$.

If three cards are drawn randomly from a pack of $52$ playing cards, then the probability of getting exactly one spade card, exactly one king, and exactly one card having a prime number is

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