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Math Help - Binomial Thm

  1. #1
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    Binomial Thm

    1.) Using the binomial theorem, prove:

    (Not sure how to do chooose in maple, so ignore the - (it's not a fraction))

    \left(\frac{n}{0}\right) + \left(\frac{n}{1}\right)\cdot 2 + \left(\frac{n}{2}\right)\cdot 2^2 + \cdots + \left(\frac{n}{n}\right)\cdot 2^n = 3^n \, \, \forall integers n \geq 0

    2.) Now prove the above giving a combinotrial proof.
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  2. #2
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    (1+2)^n = 3^n
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  3. #3
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    3^n=(1+2)^n=\sum_{i=0}^{n}\binom{n}{i}(2^{i})(1)^{  n-i}=\sum_{i=0}^{n}\binom{n}{i}(2^{i})
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