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Math Help - Combination question

  1. #1
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    Combination question

    In a poker hand of 5 cards from an ordinary deck of 52 cards, find the probability of getting a full house. Note: a full house is a hand that consists of 3 of one rank and 2 of another.

    My solution:

    \frac{13{{4}\choose{3}} * 12{{4}\choose{2}}}{{{52}\choose{5}}} +\frac{12{{4}\choose{3}} * 13{{4}\choose{2}}}{{{52}\choose{5}}}

    But, for some reason my answer is twice the answer given in the solution.
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  2. #2
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    Re: Combination question

    Quote Originally Posted by downthesun01 View Post
    In a poker hand of 5 cards from an ordinary deck of 52 cards, find the probability of getting a full house. Note: a full house is a hand that consists of 3 of one rank and 2 of another.My solution:
    \frac{13{{4}\choose{3}} * 12{{4}\choose{2}}}{{{52}\choose{5}}} +\frac{12{{4}\choose{3}} * 13{{4}\choose{2}}}{{{52}\choose{5}}}
    But, for some reason my answer is twice the answer given in the solution.
    It is because you did the same count twice. WHY?
    Thanks from downthesun01
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  3. #3
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    Re: Combination question

    I realized my mistake while in the car this morning. I multiplied both sides by the permutation of the rank pairs {AAAA},{2222},... instead of multiplying by the combination of rank pairs. My answer should have been:

    {{12\choose{2}}*\left(\frac{{{4}\choose{3}} * {{4}\choose{2}}}{{{52}\choose{5}}} +\frac{{{4}\choose{3}} * {{4}\choose{2}}}{{{52}\choose{5}}})
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