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Quadruples with an even entry

CombinatoricsHard

Count the ordered sets of $4$ non-negative integers $(a, b, c, d)$ with $a + b + c + d = 20$ and $a$ even.

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Hint

Fix the even value of the first entry, count the rest, and add up the results.

Solution

Write $a = 2 k$ with $0 \leq k \leq 10$ , so $b + c + d = 20 - 2 k$ .
For each $k$ there are $(2 22 - 2 k)$ solutions by stars and bars.
Let $m = 20 - 2 k$ , which runs over even values from $0$ to $20$ , so the terms $(2 m + 2)$ are $1 + 6 + 15 + 28 + 45 + 66 + 91 + 120 + 153 + 190 + 231$ .
Adding these terms gives the total.
The count is $946$ .

Answer: $946$