Solve $x + y + t \le 10$ 
For nonnegative integers, $x, y, t$ solve, $$x + y + t \le 10$$

This includes then: $x + y + t = 0$, ..., $x + y + t = 10$.
$x + y + t = 0$ has $1$ solution $= \binom{2}{2}$. 
$x + y + t = 1$ has $3$ solutions, $= \binom{3}{2}$
$\cdots$
$x + y + t = 10$ has: $ $ solutions: $= \binom{12}{2}$
$$= \sum_{n=2}^{12} \binom{n}{2} = \binom{13}{3}$$
Total Solutions

You can include a slack variable $s$, and then solve $$x+y+t+s=10$$
with each variable a nonnegative integer.  You can solve this with stars and bars, with solution $${10+4-1\choose 4-1}={13\choose 3}$$
