I would like to calculate the maximum number of polynomial terms given a certain number of variables and a certain degree. eg. given that the number of variables is 2 and the degree is 3, the maximum number of terms is 9:
$$x\_1^3 + x\_1^2 x\_2 + x\_1 x\_2^2 + x\_2^3+ x\_1^2 +x\_1 x\_2 + x\_2^2 + x\_1 + x\_2 + 1$$
How can I calculate this maximum number? Thanks for answering.



Number of polynomial terms for certain degree and certain number of variables
If you have $k$ variables and want degrees $1,...,n$, you get the sum $$\sum\_{i=1}^n h\_i(1,...,1)=\sum\_{i=1}^n\binom{k+i-1}{i}=\binom{k+n}{n}-1.$$ Here the $h\_i$ are the  [complete homogeneous symmetric polynomials](http://en.wikipedia.org/wiki/Complete_homogeneous_symmetric_polynomial).


