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There exist p p non-negative (0 included) integer solutions to the equation a + b + c = 3 a + b + c = 3 , q q solutions to d + e + f + g = 5 d + e + f + g = 5 and r r solutions to h + i + j + k + l = 7 h + i + j + k + l = 7 . What is p + q + r ? p + q + r? .


The answer is 396.

Elliott Macneil by Elliott Macneil , 22, United Kingdom

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1 solution

Elliott Macneil
Jun 17, 2015

This is all due to an identity which I found which states that the number of nonnegative solutions to an equation of the form a + b + c + . . . + x = n a + b + c + ... + x = n where the string of numbers is k numbers (can't find out how to do subscripts here) there are ( n + k − 1 k − 1 ) \dbinom{n + k - 1}{k - 1} (which is called 'stars and bars' in other contexts, I know). Hence, the individual values are 10, 56, and 330 respectively. Hence the answer is 396 396

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