What is the dimension of $\mathbb{C}[X,Y,Z] / \langle X^{c-2} , Y^c , Z^c \rangle$ as a vector space over $\mathbb{C}$?
I suspect this is really just a question of combinatorics as it seems my problem is equivalent to the following one:
Suppose I have $c-3$ red marbles, $c-1$ green marbles and $c-1$ blue marbles. How many distinct combinations of marbles are there as a function of $c$?
It's not too hard to do this for small $c$ but for large $c$, things get more complicated, and I'm struggling to find a formula. Any help would be appreciated.
I literally realized the solution is just the product (c-2)(c)(c). I need to count distinct monomials $\{x^i y^j z^k\}$ where $0 \leq i < c-2$, and $0 \leq j,k < c$ and so I have $c-2$ choices for $i$, and $c$ choices for $j$ and $k$.