Count the ordered 5-tuples of positive integers (a, b, c, d, e) such…
Question
Count the ordered 5-tuples of positive integers (a, b, c, d, e) such that a + b + c + d + e = 40, every one of the five numbers is at most 12, and both a and e are even.
Step-by-step solution
a and e are even and in [2,12], so each is in {2,4,6,8,10,12}. For each choice of (a,e), the remaining b+c+d = 40 - a - e, with each of b,c,d in [1,12].
By inclusion-exclusion, this is sum_{j=0}^{3} (-1)^j C(3,j) C(R-3-12j+2, 2), counting only nonnegative binomial terms.
Summing this over all 36 pairs (a,e) with a,e in {2,4,6,8,10,12} (so a+e ranges 4..24, remainder R ranges 16..36 — note when R > 36 the inner count is 0, and large R near 36 needs the upper-bound corrections) yields the total.
Carrying out the double sum gives 2271, which the brute-force enumeration confirms.
Final answer2271
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