TITLE: Dimension of set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$
QUESTION [5 upvotes]: Let, $V$ be a set of all homogeneous polynomial of degree $d$ in $n$-variables over a field $F$. Then dimension of $V_F$ is
(A) $\left(\begin{matrix}n\\d\end{matrix}\right)$
(B) $\left(\begin{matrix}d\\n\end{matrix}\right)$
(C) $\left(\begin{matrix}n+d-1\\d-1\end{matrix}\right)$
(D) $\left(\begin{matrix}n+d\\d\end{matrix}\right)$.

I tried through example for $n=2,3,4$ and $d=2,3,4$ and try to generalize the formula.
For $n=3$ and $d=3$  we get the basis of $V$ is $\{x^3,y^3,z^3,xy^2,yx^2,yz^2,zy^2,xz^2,zx^2,xyz\}$. So $dim(V)=10$. Similarly for $n=3$ and $d=2$ we get $dim(V)=6$. Also for $n=2$ and $d=2$ $dim(V)=3$. But I could not generalize these for arbitrary $n$ and $d$.
Suggest to find the general formula or any other way to determine it directly..

REPLY [4 votes]: The standard basis consists of all monomials $x_1^{a_1}x_2^{a_2}\cdots x_n^{a_n}$, where the $a_i$ are non-negative integers such that $a_1+\cdots +a_n=d$.
By Stars and Bars the number of such $n$-tuples $(a_1,\dots,a_n)$ is $\binom{n+d-1}{n-1}$ or equivalently $\binom{n+d-1}{d}$.
There are also many questions/answers on MSE that use the Stars and Bars argument.
Remark: None of the suggested answers is correct!