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Math Help - [SOLVED] Card Combinations

  1. #1
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    [SOLVED] Card Combinations

    If a bridge hand consists of 13 cards, how many bridge hands include 5 cards of one suit, 6 cards of a second suit, and 2 cards of another suit?

    Since we don't care what the cards are and just what their suit is would the answer be 4P3 since the order of the suits matters since there are a different number of each cards of a particular suit?
    Last edited by zerobladex; February 19th 2009 at 02:02 PM.
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  2. #2
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    Hello, zerobladex!

    If a bridge hand consists of 13 cards, how many bridge hands include:
    5 cards of one suit, 6 cards of a second suit, and 2 cards of another suit?

    First, select 3 of the 4 suits and place them in an order.

    [The first suit is the 5-card set, the second is the 6-card set, the third the 2-card set.]
    . . There are: . _4P_3 ways.


    Then we will choose:

    . . \begin{array}{cc}\text{5 of the 1st suit:} &_{13}C_5\text{ ways.} \\ \\[-4mm]<br />
\text{6 of the 2nd suit:} & _{13}C_6 \text{ ways.} \\ \\[-4mm]<br />
\text{2 of the 3rd suit:} & _{13}C _2 \text{  ways.}<br />
\end{array}


    Therefore, there are: . \left(_4P_3\right)\left(_{13}C_5\right)\left(_{13}  C_6\right)\left(_{13}C_2\right)

    . . . . . . . . . . . . . . = \;(24)(1,\!287)(1,\!716)(78)

    . . . . . . . . . . . . . . = \;4,\!134,\!297,\!024\text{  hands.}

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