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Math Help - Prove that the binomial coefficient is always a natural number?

  1. #1
    Member mfetch22's Avatar
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    Prove that the binomial coefficient is always a natural number?

    So say we have the binomial coefficient of n (on the top) and k ( on the bottom) and this text book says to prove by induction that the binomial coefficient is always a natural number, but how can I prove something by indunction when there are two variables involved? Do I only need to use induction on one of the variables? And also, how would I go about actually "starting up" my induction argument? Thanks in advance.
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    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by mfetch22 View Post
    So say we have the binomial coefficient of n (on the top) and k ( on the bottom) and this text book says to prove by induction that the binomial coefficient is always a natural number, but how can I prove something by indunction when there are two variables involved? Do I only need to use induction on one of the variables? And also, how would I go about actually "starting up" my induction argument? Thanks in advance.
    Combinatorial proof:

    It is the number of combination of k elements from group of n (natural number) = n choose k.
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    A.S.Z's combinatorial proof is correct, but it sounds like you are supposed to provide an inductive proof.

    Do you know this identity?

    \binom{n}{m} + \binom{n}{m+1} = \binom{n+1}{m+1}

    If so, you might use it as the basis of your proof.
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