Binomial Theorem

**Zero266** · January 1st 2009, 01:06 PM

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 )

**Plato** · January 1st 2009, 01:24 PM

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 x’s and two y’s.
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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