How many positive whole numbers below 1000 have digits that add up to 10?

One-digit: none. Two-digit: 19, 28, …, 91 — 9 numbers. Three-digit: $a+b+c=10$ with a ≥ 1 and each digit ≤ 9: there are $\left(\genfrac{}{}{0ex}{}{11}{2}\right)=55$ ways with a ≥ 1, minus 1 $(a=10)=54$. Total 9 + 54 = 63.