We know that the number of terms in a univariate polynomial of degree n is n+1. But what about if there are multiple variables: for eg: for variables $x,y$ polynomial of degree 2 will have:
$1+x+y+xy+x^2+y^2$
(By degree, here i mean the combined power of $x$ and $y$)
To find out the number of temrs in a polynomial with $r$ variables of degree $n$.
I found out the number of terms of each power (combined power) up-to $n$ and summed them.
The number of terms of power $k$ will be equivalent to the number of non-negative integral solutions of :
$p_1 + p_2 + ... p_r = k$
which is equal to $ k+r-1 \choose r-1$.
So i figured the number of terms in the polynomial will be:
$\sum_{k=0}^{n} {k+r-1 \choose r-1}$
Is there a way to simplify the above sum into a single term (like $\sum_{r=0}^{n} {n \choose r} = 2^n$)?
Thanks for the help.