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Math Help - Proofs im having trouble with

  1. #1
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    Proofs im having trouble with

    Prove Pascals Formula by using the fact that nCk = n!/(k!(n-k)!)

    Prove that k*(nCk) = n*(n-1)C(k-1) by using the same fact from above.

    Use the lattice walk idea to prove that nCk = nC(n-k).

    Use the lattice walk idea to prove nC0 + nC1 + ... + nCn = 2^n



    If anyone can show any of these, I would greatly appreciate it. Thanks!
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  2. #2
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    Quote Originally Posted by jzellt View Post
    Prove Pascals Formula by using the fact that nCk = n!/(k!(n-k)!)

    Prove that k*(nCk) = n*(n-1)C(k-1) by using the same fact from above.

    Use the lattice walk idea to prove that nCk = nC(n-k).

    Use the lattice walk idea to prove nC0 + nC1 + ... + nCn = 2^n



    If anyone can show any of these, I would greatly appreciate it. Thanks!


    I use \binom{n}{k} instead of nCk , so:

    \displaystyle{k\binom{n}{k}=k\frac{n!}{k!(n-k)!}=\frac{n!}{(k-1)!(n-k)!} , since \displaystyle{m(m-1)!=m!\Longleftrightarrow \frac{m}{m!}=\frac{1}{(m-1)!}}

    Now show that the above equals \displaystyle{n\binom{n-1}{k-1}}

    I don't know what the path walk idea is, but the last equality follows at once

    from Newton's Binom: \displaystyle{(a+b)^n=\sum\limits^n_{k=0}\binom{n}  {k}a^{n-k}b^k}

    Tonio
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