Relation between number of possible arrangements of $x's$ and $y's$ and the coefficient of the $(r+1)^{th}$ term

Question

Consider the expansion of $(x+y)^n$, $$ (x+y)^n = \sum_{r=0}^n\binom{n}r x^{n-r}y^r$$ Here the coefficient of the $(r+1)^{th}$ term is the number of ways in which $(n-r)\ x$'s and $ r \ y$'s can be arranged. It is given by, $$ \frac{n!}{(n-r)!r!}$$

Why is the number of possible arrangements linked to the coefficient of the $(r+1)^{th}$ term?

Answer 1

If you see it as $(x+y)^n = (x+y)...(x+y)$, then coefficient of $x^{(n-r)}y^{(r)}$ is the number of choices of $(n-r)$, $x$ and $(r)$, $y$ from $n$ parentheses. that likes an arrangement of these.