 Solution

ID: 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 of

d_{1} + d_{2} + \dots + d_{6} = 5.

By stars and bars, there are

(6 - 1 5 + 6 - 1) = (5 10)

. The all-zero string is not counted because its digit sum is zero, so every counted integer is positive. Thus the answer is $252$ .

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