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

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- Solution 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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