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Math Help - Hard proof.

  1. #1
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    Hard proof.

    Prove the following identity:

    (m choose k) = sum from k = 0 to n of (n choose k) (m - n choose n - k).

    Thanks.
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  2. #2
    ynj
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    Quote Originally Posted by cgiulz View Post
    Prove the following identity:

    (m choose k) = sum from k = 0 to n of (n choose k) (m - n choose n - k).

    Thanks.
    Imagine there are two piles of balls. One has n balls, another one has m-n balls. Now you want to choose k balls from them. So you must decide how many balls you choose from the first pile. Suppose you choose k balls, then there would be \left(\begin{array}{cc}n\\k\end{array}\right)\left  (\begin{array}{cc}m-n\\n-k\end{array}\right)kinds of method. Then calculate the sum for all the k.
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