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Annual Harvard-MIT Mathematics Tournament

Combinatorics Difficulty 5.0 AIME Prove it United States

Problem:

Determine the number of 8-tuples of nonnegative integers $\left(a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}\right)$ satisfying $0 \leq a_{k} \leq k$ , for each $k=1,2,3,4$ , and
$a_{1} + a_{2} + a_{3} + a_{4} + 2 b_{1} + 3 b_{2} + 4 b_{3} + 5 b_{4} = 19.$

Solution

Answer: 1540 Same as Combinatorics Test problem 10.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.