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Math Help - Combination

  1. #1
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    Unhappy Combination

    Prove that (n+1)C4=((nc2)c2)/3 ,where n>=4
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  2. #2
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    Re: Combination

    \binom{n}{2} = \frac{n(n-1)}{2} = \frac{n^2 - n}{2}. Also, \binom{\binom{n}{2}}{2} = \frac{\frac{n^2 - n}{2} \frac{n^2 - n - 2}{2}}{2} = \frac{(n^2 - n)(n^2 - n - 2)}{8}. So you can simplify the RHS. Then use algebra.
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  3. #3
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    Re: Combination

    Hello, Swarnav!

    \text{Prove: }\:{n+1\choose4} \:=\:\frac{1}{3}{{n\choose2}\choose2}\;\text{where }n \ge 4

    \text{RHS: }\:\frac{1}{3}\cdot{\frac{n(n-1)}{2\cdot1} \choose 2} \;=\;\frac{1}{3}\cdot\frac{\left(\frac{n(n-1)}{2}\right)\left(\frac{n(n-1)}{2} - 1\right)}{2\cdot 1} \;=\;\frac{1}{3}\cdot\frac{\left(\frac{n(n-1)}{2}\right)\left(\frac{n(n-1)-2}{2}\right)}{2\cdot 1}

    . . . =\;\frac{1}{3}\cdot\frac{n(n-1)(n^2-n-2)}{4\cdot2\cdot1} \;=\; \frac{1}{3}\cdot\frac{n(n-1)(n+1)(n-2)}{4\cdot2\cdot1}

    . . . =\;\frac{(n+1)n(n-1)(n-2)}{4\cdot3\cdot2\cdot1} \;=\;{n+1\choose4} \;=\;\text{LHS}
    Last edited by Soroban; July 14th 2012 at 08:55 AM.
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