We saw the following theorem in class:
If $n \in \mathbb{N}$ and $z_1, \dots, z_m \in \mathbb{C}$ we have:
$ (z_1 + \dots + z_m)^n= \sum_{k_1+\dots+k_m=n} \binom{n}{k_1, \dots, k_m} z_1^{k_1} \dots z_m^{k_m} $
And I tried to understand the combinatorial proof of this theorem, which is as following:
We first consider a term in the expansion of $(z_1 + \dots + z_m)^n$. This must be of the form $\alpha \prod_{i=1}^k x_i^{\beta_i}$ for some integer $\alpha $ and some integers $\beta_i$.
Now it says that: Since each term must come from choosing one summand from $z_1 + \dots + z_m$ we must have $\beta_1+\beta_2+ \dots + \beta_k =n$. That what confuses me here is, what does it mean when one says that each term must come from choosing one summand? For example $(x_1 +x_2)^2 = x_1^2+2x_1x_2+x_2^2$. Then the term $2x_1x_2$ doesn't come from choosing one summand, right? We choose $x_1$ and $x_2$.
Further they say from that it follows that $\beta_1+\beta_2+ \dots + \beta_k =n$. I see that this is true, but I don't see why this has to follow from the statement "choosing one summand from $z_1 + \dots + z_m$".
In the end the argument is like: THe number of different ways that we can get this term will be the number of ways to choose $\beta_1$ copies of $x_1$, $\beta_2$ copies of $x_2$ and so on. How can I conclude from that the claim?
For example, $(x+y)^2=x^2+2xy+y^2$ comes from choosing $$ \begin{array}{c|c|c} &(x+y)&(x+y)\\ \hline x^2&x&x\\ xy&x&y\\ yx&y&x\\ y^2&y&y\\ \end{array} $$ and $(x+y)^3=x^3+3x^2y+3y^2x+y^3$ comes from choosing $$ \begin{array}{c|c|c|c} &(x+y)&(x+y)&(x+y)\\ \hline x^3&x&x&x\\ x^2y&x&x&y\\ x^2y&x&y&x\\ x^2y&y&x&x\\ xy^2&x&y&y\\ xy^2&y&x&y\\ xy^2&y&y&x\\ y^3&y&y&y\\ \end{array} $$