Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics and probability resit / 1 / d / Solution

Pad every integer from $0$ through $999999$ with leading zeros to obtain a unique six-digit decimal string. If its digits sum to $5$, each digit is automatically at most $5$, so the usual digit upper bound $9$ adds no restriction. We therefore count the nonnegative integer solutions ofBy stars and bars, there are $\left(\genfrac{}{}{0ex}{}{5+6-1}{6-1}\right)=\left(\genfrac{}{}{0ex}{}{10}{5}\right)$. The all-zero string is not counted because its digit sum is zero, so every counted integer is positive. Thus the answer is $252$.