Three-digit codes with nonzero digits summing to 10 or less

A safe opens with a code of three digits. No digit is $0$, and the three digits together add up to at most $10$. How many codes are possible?

$120$. Add a slack variable: $d_1+d_2+d_3+s=10$ with $d_i\ge1$, $s\ge0$. Substituting $d_i'=d_i-1$ gives four non-negative parts summing to $7$, so $\binom{10}{3}=120$ (the cap of $9$ per digit never binds).