<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] Can use 'stars and bars'/combination 'with repetition':
number of positive integer solutions to $x+y+z=15$ is the same as the number of ways you can place $15$ identical objects in $3$ containers or ${(n-1)+r}\choose r$ with $n=3$ and $r=15$.  gives a total of ${17\choose 15}=136$.  To add restrictions simply subtract all applicable combinations 'with repetition':  
for $1)$ if $x$ is $0$ the fifteen objects must be arranged exclusively in the other $2$ containers: 
$n=2,r=15$: ${{(n-1)+r}\choose r}={16\choose 15}$
if $x\ge 7$:  $r=15-7=8$,$n=3$: ${{(n-1)+r}\choose r}={10\choose 8}$
so solution is: ${17\choose 15}-{16\choose 15}-{10\choose 8}=136-16-45=75$
$2)$ and $3)$ can be solved similarly.
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