Binomial theorem combinatorics derivation We can split any term $(x+y)^n$ into a choice tree with $2^n$ path choices.  Also, we can represent each factor in terms of $x^ky^{n-k}$
It is logical to deduce we can express this with summation notation: 
$$\sum_{k=0}^{n}{cx^ky^{n-k}}$$
where c is the coefficient of the term.  Everything makes sense to me, except how we can find c.  Apparently, we can define c as $$\binom{n}{k}$$
But this is confusing for me, and the only definition I can find is a very intense mathematical one:
The coefficient of $x^ky^{n-k}$ for a particular k is just the number of ways to choose k factors of y from the n factors of (x+y), with factors of x coming from the remaining (n−k) factors. The number of ways to choose k objects from a collection of n objects (without replacement, order not important) is just $\binom{n}{k}$.
I understand everything logically, and I've been trying to figure this out all day.  I'm not new to combinatorics as I've taken a few statistics classes.  But these were the basics, such as the number of ways to chose 2 fruits from a set of 3 fruits (say an apple, orange, banana). 
Firstly, there are n+1 factors, and $2^n$ paths.  So I just don't get the logic behind all of this, even though it does work.

You can go factor by factor through the product $(x+y)\cdots(x+y)$, at each one choosing whether you want to pick the $x$ or the $y$. Or, you can pick which of the $n$ factors you want to pick the $x$s out of all at once, which automatically determines which the $y$s must be. This specifies the same information, really.
