If we consider a multivariate polynomial of $n$ variables with degree $N$, how do we show that the maximal number of monomials is $$\binom{N+n}{n} = \frac{(N+n)!}{n!\, N!}\quad ?$$
The hint our teacher gave was that the number of monomials is the number of independent coefficients. So how many independent coefficients does a multivariate polynomial have (at most)?
Thank you.
Let $x_1, \ldots, x_n$ be the $n$ variables. The typical term of the polynomial has the form $$Cx_1^{k_1}x_2^{k_2}\cdots x_n^{k_n},$$ where each $k_i \geq 0$ and $$k_1 + \cdots + k_n \leq N.$$ Thus, the problem reduces to finding the number of $n$-tuples $(k_1, \ldots, k_n)$ satisfying this constraint. There are $\binom{N+n}{n}$ such $n$-tuples. This is a combinatorial exercise within the main exercise.
The main idea is to associate to each $n$-tuple an $n$-element subset of $\{1, \ldots, N + n\}$, namely $\{k_1 + 1, k_1 + k_2 + 2, \ldots, k_1 + k_2 + \cdots + k_n + n\}$. Once you've shown this is a bijection, the result follows.
