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Math Help - How many ways can this type of 12 card hand be chosen?

  1. #1
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    How many ways can this type of 12 card hand be chosen?

    How many ways can 12 cards be chosen from a standard deck of 52 cards such that the same number of cards from each suit are in the 12?

    The way I saw it was that initially, you had to chose 1 of the 4 suits, i.e., 4\choose{1}. Then within the chosen suit you needed to choose 3 cards, i.e., 13\choose{3}, then repeat the process with the remaining 3 suits, so my solution was:

    4\choose{1} 13 \choose{3} 3\choose{1 } 13\choose{3} 2\choose{1} 13\choose{3} 1\choose{1} 13 \choose{3}

    The answer provided says:

    13 \choose{3} 13 \choose{3} 13 \choose{3} 13 \choose{3}

    Why is the choice of suit not included?
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  2. #2
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    Re: How many ways can this type of 12 card hand be chosen?

    Quote Originally Posted by terrorsquid View Post
    How many ways can 12 cards be chosen from a standard deck of 52 cards such that the same number of cards from each suit are in the 12?

    The way I saw it was that initially, you had to chose 1 of the 4 suits, i.e., 4\choose{1}. Then within the chosen suit you needed to choose 3 cards, i.e., 13\choose{3}, then repeat the process with the remaining 3 suits, so my solution was:

    4\choose{1} 13 \choose{3} 3\choose{1 } 13\choose{3} 2\choose{1} 13\choose{3} 1\choose{1} 13 \choose{3}

    The answer provided says:

    13 \choose{3} 13 \choose{3} 13 \choose{3} 13 \choose{3}

    Why is the choice of suit not included?
    13 choose 3 indicates you are picking one suit. You are doing that 4 times i.e. all 4 suits are being considered.
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  3. #3
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    Re: How many ways can this type of 12 card hand be chosen?

    I realize this is a few days old and dwsmith covered it quite succinctly. However, just in case: You are mixing up probability with choices. There is no probability in this question at all. Its 100% likely you are going to get 12 cards, in which each suit has 3 cards. Imagine the whole deck is laid out in front of you face up. You can pick any 3 hearts you want, any 3 spades you want ,etc.

    How many different ways can you do that? dwsmith has the answer right above.
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