<s>[INST] Find the number of non-negative integer solutions to $x+y+z=11$ How do I find the number of nonnegative integer solutions to $x+y+z=11$ provided that $x\leq 3, y\leq 4, z\leq 6$ using the sum rule (counting)?
I know the answer is 6, but I'm having difficulty understanding why.
 [/INST] One way is to expand $$(1+x+x^2+x^3)(1+x+x^2+x^3+x^4)(1+x+x^2+x^3+x^4+x^5+x^6)$$ and get the coefficient of $x^{11}.$
</s>