Dimension of polynomial ring

Question

Suppose I have a polynomial ring $k[x_0,\cdots,x_n]$ and let $k[x_0,\cdots,x_n]_m$ denote the homogeneous part with degree $m$. How can I see that this $k[x_0,\cdots,x_n]_m$ has dimension $\binom{m+n}{m}$? Or more exactly, why does this give the Krull dimension?

Answer 1

Hint:

If you denote $k_i$ the exponent of $xi$in such a monomial, it is the number of solutions in natural numbers of the equation $$k_0+k_1+\dots+ k_n=m.$$