How to compute $ \mathrm{dim}_{ \mathbb{C} } \big( \mathbb{C} [X, Y] / I^n ) $? 
Can you show me how to compute $ \mathrm{dim}_{ \mathbb{C} } \big( \mathbb{C} [X, Y] / I^n  \big) $ where $ I = (X,Y) \subset \mathbb{C} [X,Y] $ ? 

I have to find : $ \mathrm{dim}_{ \mathbb{C} } \big( \mathbb{C} [X, Y] / I^n ) = 1 + 2 + \dots + n = \dfrac{n(n+1)}{2} $, but i don't know how.  
Thank you in advance.

Choose a basis of monomials $x^ky^l$ with $k$ and $l$ non-negative integers. Obviously you need only consider those with $k+l<n$. 
Now let $M:=k+l$. There are $M+1$ ways to write an integer $M$ as a sum of non-negative integers.
Hence in our case there are $1$ monomial of degree $0$, $2$ monomials of degree $1$, ..., $n$ monomials of degree $n-1$.
