SOLUTION: You have a deck of 52 playing cards. What is the probability that a hand of 5 cards dealt randomly will contain 3 aces

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Question 1058578: You have a deck of 52 playing cards. What is the probability that a hand of 5 cards dealt randomly will contain 3 aces
Answer by solve_for_x(190) About Me  (Show Source):
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There are 4 Aces in the deck, and 52 - 4 = 48 cards which are not Aces.

A five card hand that contains 3 Aces will contain 2 non-Aces.

The number of possible 5-card hands with 3 Aces is:

C(4, 3) * C(48, 2) = %284%21%2F3%21%284-3%29%21%29%2848%21%2F2%21%2848-2%29%21%29 = 4 * 1128 = 4512 hands

The number of possible 5-card hands of any combination is:

C(52, 5) = 52%21+%2F+5%21%2852+-+5%29%21 = 2598960 hands

The probability of a hand with 3 Aces is then:

P(3 Aces) = 4512 / 2598960 = 94 / 54145 ≈ 0.001736