Combinatorics Problem

blue_soda025

Suppose you play a game of cards in which only four cards are dealt from a standard deck of 52 cards. How many ways are there to obtain three of a kind? (3 cards of the same rank and 1 card of a different rank, for example 3 tens and 1 queen.)

Could someone help me with how to do this problem? I tried doing ₄C₃ x 13 x ₅₂C₁, which was obviously wrong. :/ Any help would be appreciated.

KingOfTwilight

You can choose first card in 52 different ways. You can choose the next two (same of a kind as the first one) in 3 * 2 ways. And the last card, in 49 different ways.

xanthym

The standard 52 cards contain 13 different ranks of 4 cards each.
For any given rank, there are C(4,3) combinations of 3 cards chosen from the rank's 4 cards. Since there are 13 different ranks, a total of {13*C(4,3)} possible combinations of {3 cards from the same rank} exist. Finally, there remain {(52 - 4) = 48} cards in the 12 other (different) ranks from which to choose the final card. Hence:
{Total Combinations of "3-of-a-Kind" from std 52 Cards} = {13*C(4,3)}*(48) = (2496)


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