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Thread: Binomial Coefficients Formula

  1. #1
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    Binomial Coefficients Formula

    Proof that $\displaystyle {n\choose{k}} = \frac{n!}{k!(n-k)!} = \frac{(n-k+1)(n-k+2)\cdot\cdot\cdot(n-1)n}{2\cdot{3}\cdot\cdot\cdot(k-1)k}$
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  2. #2
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    Quote Originally Posted by Mozart View Post
    Proof that $\displaystyle {n\choose{k}} = \frac{n!}{k!(n-k)!} = \frac{(n-k+1)(n-k+2)\cdot\cdot\cdot(n-1)n}{2\cdot{3}\cdot\cdot\cdot(k-1)k}$
    You can write the $\displaystyle n!$ in the numerator as

    $\displaystyle 2\cdot 3 \cdot 4 \cdot \dots \cdot (n - k)\cdot(n - k + 1)\cdot(n - k + 2)\cdot \dots (n - 1)\cdot n$

    The $\displaystyle (n - k)!$ in the denominator is

    $\displaystyle 2 \cdot 3 \cdot 4 \cdot \dots \cdot (n - k)$.

    Can you see anything that cancels?


    Also, $\displaystyle k! = 2 \cdot 3 \cdot 4 \cdot \dots \cdot k$.
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