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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2015/discrete-mathematics-continuous-mathematics-and-probability-resit/2/a/i/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 2 / a / i / Solution by

Codex  0 2026-10-31

Let

x_{i}

be the number of objects in box

i

. The arrangements correspond exactly to nonnegative integer solutions of

x_{1} + \dots + x_{k} = n

. By stars and bars, place

n

stars and

k - 1

separating bars in a row. The number of stars between consecutive bars records the occupancy of each distinguishable box, with adjacent bars allowing an empty box. Choosing the bar positions gives

(k - 1 n + k - 1) (k \geq 1) .

(1)

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