How many terms are there?

Question

Consider the expansion: $$(x^2+x^3+x^4+x^5+x^6+x^7+x^8)^4$$ How many different terms are there in this expansion? Actually the problem is I can not apply the multinomial theorem because it comes same factors from the product of different terms, I mean for example $x^2 × x^5$ gives $x^7$ and also $x^3 × x^4$ gives $x^7$. How can I control to count all distinct terms only once? How can I generalize this example for any number of $x$'s and any positive integer powers?

Answer 1

All combinations of terms $x^ix^jx^kx^l$ where $i,j,k,l = 2,3,4,5,6,7,8$. As all combinations can and will be done these can and will be as low as $8$ and as high as $32$.

The real question is what will the coefficients be. And the answer to that is that the coefficient of $x^m$ will be the number of ways to solve $i+j+k+l = m$ with $2,3,4,5,6,7,8$. Which is the number of ways to solve $i+j+k+l = m -8$ with $0....6$.