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Math Help - Binomial Theorem

  1. #1
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    Binomial Theorem

    Hi! Can someone give me a derivation of the binomial theorem for dummies please, thanks! I'm looking for something intuitive and insightful because I just can't follow my textbook!

    (reminder: the theorem is an expansion for (x+y)^n )
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  2. #2
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    Take a simple example: \left( {x + y} \right)^6  = \left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right).
    To expand that product we take either x or y from each of the six factors.
    In order to get x^6 we must take an x from each factor. That can be done in only one way.

    But to get a term containing x^4y^2, we choose four xs and two ys.
    We could do it as xxyxxy or yxxyxx for example. That can be done is 6 \choose 4 ways.

    In general, \left( {x + y} \right)^6  = \sum\limits_{k = 0}^6 {{6\choose k}x^{6 - k} y^k } .

    Just think about the number of ways to choose the terms from the n factors.
    \left( {x + y} \right)^n  = \sum\limits_{k = 0}^n {{n\choose k} x^{n - k} y^k } .
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