<s>[INST] How many nonnegative integer solutions does the following equation have: $x + y + z = 15$ 
How many nonnegative integer solutions does the following equation
  have: $x + y + z = 15$
(1) if $1 \le x \le 6$
(2) if $x \ge 2$ and $y \le 3$
(3) if $x \ge 3$, $y \ge 2$, and $1 \le z \le 3$

I solved the first question (which is C(3+11-1,11)). Still very stuck on the next two
 [/INST] There are $75$ ways to do it. Using your procedure we get $\binom{17}{15}-\binom{10}{8}-\binom{16}{15}$. The last subtraction is to remove the cases where $x=0$.  (I also did it another way, again getting $75$. Makes $75$ twice as true.)
Added: For the second question, we want the number of solutions of $x+y+z=13$, with the restriction that $y\le 3$.
Count the number of solutions of $x+y+z=13$. Subtract the number of solutions of $x+y+z=9$. That's because to count the number of "bad" solutions of $x+y+z=13$, bad because $y\ge 4$, we give $4$ to $y$ and distribute the remaining $9$ between $x$, $y$, and $z$.
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