Count the ordered sets of 4 integers (x1, x2, x3, x4) with 0 ≤ xi ≤ 5 for every i and x1 + x2 + x3 + x4 = 15.

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Bounded integer quadruples

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Count the ordered sets of $4$ integers $(x 1, x 2, x 3, x 4)$ with $0 \leq x i \leq 5$ for every $i$ and $x 1 + x 2 + x 3 + x 4 = 15$ .

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Hint

Reflect each variable around the middle of its range to make the total small.

Solution

Substitute $y i = 5 - x i$ , so $0 \leq y i \leq 5$ and $y 1 + y 2 + y 3 + y 4 = 5$ .
The upper bound $y i \leq 5$ holds automatically because the total is $5$ .
So we count non-negative solutions, which gives $(3 5 + 3) = (3 8)$ .
Compute $(3 8) = 56$ .
The count is $56$ .

Answer: $56$