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How many non-negative integer solutions are there to x1+x2+x3+x4+x5 = 60 (a) where x1 <= 17...

How many non-negative integer solutions are there to
x1+x2+x3+x4+x5 = 60
(a) where x1 <= 17 and x2 <= 17
(b) where x1 <= 17 and x2 <= 17 and x3 <= 17 and x4 <= 17

Side question: is there any reason why, for (a), that we can't just give x1 and x2 both 18 "stars" (from the sticks and stars representation of the problem) and then calculate the number of ways to distribute the remaining 60 - (18 * 2) = 24 stars and subtract that from the total number of ways to distribute without any constraint? i.e. answer = (64 choose 4) - (28 choose 4)

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