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Math Help - combinations: possibly really easy?

  1. #1
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    combinations: possibly really easy?

    how many different ways could you choose 6 cards from a standard deck of 52 cards, if you must have atleast one card from each suit, and order does not matter.

    I got 8682544, is this correct?
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  2. #2
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    Quote Originally Posted by white_cap View Post
    how many different ways could you choose 6 cards from a standard deck of 52 cards, if you must have atleast one card from each suit, and order does not matter. I got 8682544, is this correct?
    Yes that seems to be correct.
    \sum\limits_{k = 0}^3 {\left( { - 1} \right)^k \left( {\begin{array}{c}<br />
   4  \\<br />
   k  \\<br />
\end{array}} \right)\left( {\begin{array}{c}<br />
   {52 - 13 \cdot k}  \\<br />
   6  \\<br />
\end{array}} \right)}
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  3. #3
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    Hello, white_cap!

    How many different ways could you choose 6 cards from a standard deck of 52 cards,
    if you must have at least one card from each suit, and order does not matter.

    I got 8682544, is this correct? . . . . I got the same answer!
    I "talked" my way through it . . .


    We choose six cards, and all four suits are represented.

    There are two distributions of suits.


    (1) There is 1 of the first suit, 1 of the second suit, 1 of the third suit,
    . . and 3 of the fourth suit: . [1,\,1,\,1,\,3]

    There are: .4 choices for the "triple" suit.
    Then there are: . {13\choose1}{13\choose1}{13\choose1}{13\choose3} \:=\:628,342 ways to choose the cards.

    Hence: there are: . 4 \times 628,342 \:=\:{\bf 2,513,368} ways to draw [1,\,1,\,1,\,3]



    (2) There is 1 of the first suit, 1 of the second suit, 2 of the third suit,
    . . and 2 of the fourth suit: . [1,\,1,\,2,\,2]

    There are: . {4\choose2,2} = 6 to select the suits.
    Then there are: . {13\choose1}{13\choose1}{13\choose2}{13\choose2} \:=\:1,028,196 ways to choose the cards.

    Hence, there are: . 6 \times 1,028,196 \:=\:{\bf6,169,176} ways to draw [1,\,1,\,2,\,2,]



    Therefore, there is a total of: . 2,513,368 + 6,169,176 \;=\;{\bf{\color{blue}8,682,544}} ways.

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