In a standard deck of 52 cards how many different two of a kind hands (2 cards of one rank, 2 of another, and 1 of a third) are possible?

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  • 4 weeks ago
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    This hand is more accurately called "two pairs". (Two of a kind is a pair and then 3 non-matching cards).

    First pick the two ranks that will be part of the pairs (e.g. Kings and 7s)

    C(13,2) = (13 * 12) / 2 = 78 ways

    Pick the two suits for the higher card:

    C(4,2) = (4 * 3) / 2 = 6 ways

    Pick the two suits for the lower card:

    C(4,2) = (4 * 3) / 2 = 6 ways

    From the remaining 11 ranks, pick the rank of the single card:

    C(11,1) = 11 ways

    Pick the suit for that single card:

    C(4,1) = 4 ways

    So the total ways to get two pairs are:

    78 * 6 * 6 * 11 * 4

    = 123,552 ways 

    P.S. You'll notice this matches with the frequency column for two pairs in the following Wikipedia article.

  • 4 weeks ago

    The first card can be any of the 52 cards.

    The second card must be one of the 3 cards that makes it a pair.

    The third card must be one of the 48 cards left that does't make it a three-of-a-kind.

    The fourth card must be one of the 3 cards that makes it a two-of-a-kind.

    The fifth card must be one of the 44 cards left that doens't make it a full house.

    Since your pairs would have A♥ and A♦ as one hand and A♦ and A♥ as a different pair we have to divide this count by 2 for each pair to get rid of the duplicates.

    Finally, the two pairs can be interchanged So if you have AATTK and TTAAK this would be considered the same hand, so we'll divide this by 2 one more time to get rid of this set of duplicates.

    Multiply everything together to get:

    52 * 3 * 48 * 3 * 44 * (1/2) * (1/2) * (1/2) = 123,552 possible poker hands that results in a two of a kind.

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