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 )
Take a simple example: $\displaystyle \left( {x + y} \right)^6 = \left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)\left( {x + y} \right)$.
To expand that product we take either x or y from each of the six factors.
In order to get $\displaystyle x^6$ we must take an x from each factor. That can be done in only one way.
But to get a term containing $\displaystyle x^4y^2$, we choose four x’s and two y’s.
We could do it as xxyxxy or yxxyxx for example. That can be done is $\displaystyle 6 \choose 4$ ways.
In general, $\displaystyle \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.
$\displaystyle \left( {x + y} \right)^n = \sum\limits_{k = 0}^n {{n\choose k} x^{n - k} y^k } $.