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Math Help - Different ways of Counting ....

  1. #1
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    Different ways of Counting ....

    (a) Let n be a nonnegative integer. Use the identity

    (1 + x)^n(1 + x)^n = (1 + x)^ (2n)

    to show that  \sum_{n=0}^{n}\binom{n}{k}^2 =\binom{2n}{n} (1)

    (b) Prove (1) again by counting, in two different ways, the number of ways
    of choosing n people from a set of n girls and n boys.
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  2. #2
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    Quote Originally Posted by shmounal View Post
    (a) Let n be a nonnegative integer. Use the identity

    (1 + x)^n(1 + x)^n = (1 + x)^ (2n)

    to show that  \sum_{n=0}^{n}\binom{n}{k}^2 =\binom{2n}{n} (1)


    Compare the coefficient of x^2 in both expressions above.

    Tonio


    (b) Prove (1) again by counting, in two different ways, the number of ways
    of choosing n people from a set of n girls and n boys.
    .
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