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Math Help - Binomial Coefficient Proof

  1. #1
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    Binomial Coefficient Proof

    Prove that, for all real numbers r and all integers k and m,

    C(r,m)C(m,k)=C(r,k)C(r-k,m-k)

    Where C(r,m) is r choose m.

    I'm terrible at these proofs because I never know which identity is needed to start.

    Any help?
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  2. #2
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    Easier to think of this through combinatorics.

    Left hand side:
    r = number of people
    m= # of people being chosen for some committee from the group of r people.
    k = Choosing k people out of the committee of m people for some special role (say senators).

    Right hand side
    Choosing k people who will be senators and in a committee. Then you designate the remaining empty committee spots (m-k) to the left over people (r-k)
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  3. #3
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    Quote Originally Posted by coolhandluke View Post
    Prove that, for all real numbers r and all integers k and m,

    C(r,m)C(m,k)=C(r,k)C(r-k,m-k)

    Where C(r,m) is r choose m.

    I'm terrible at these proofs because I never know which identity is needed to start.

    Any help?
    An algebraic proof is also easy.

    \binom{r}{m} \binom{m}{k} = \frac{r!}{m!(r-m)!} \frac{m!}{k! (m-k)!} ...(1)

    \binom{r}{k} \binom{r-k}{m-k} = \frac{r!}{k! (r-k)!} \frac{(r-k)!}{(m-k)! (r-m)!} ...(2)

    Now compare (1) and (2).
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