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.