Counting numbers up to 100000 whose digits add to nine

How many positive integers not exceeding $100{,}000$ have a digit sum of exactly $9$? (For instance, $81$, $144$ and $13212$ are such numbers.)

$715$. $100{,}000$ itself has digit sum $1$, so pad every candidate to five digits with leading zeros. We then count solutions of $d_1+\cdots+d_5=9$ in non-negative integers; no digit can exceed $9$ anyway. Stars and bars gives $\binom{13}{4}=715$.