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Thread: [SOLVED] urgent!! prove using binomial throrem

  1. #1
    bdou
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    Exclamation [SOLVED] urgent!! prove using binomial throrem

    hey! i have a question from my number theory worksheet.

    prove that nC0+nC1+nC2+...+nCn=2^n and
    nC0-nC1+nC2-...+(-1)^n(nCn)=0

    it hints to use the binomial theorem
    any help would be amazing!
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  2. #2
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    $\displaystyle \begin{array}{l}
    \left( {x + y} \right)^n = \sum\limits_{k = 0}^n {_n C_k } x^k y^{n - k} \\
    \mbox{Let}\quad x = 1\quad \& \quad y = 1 \\
    \end{array}
    $
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  3. #3
    MHF Contributor red_dog's Avatar
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    For the second one:
    $\displaystyle \displaystyle (x-y)^n=\sum_{k=0}^n(-1)^kC_n^kx^{n-k}b^k$ and set $\displaystyle x=y=1$
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