The coefficient of \(x^{12}\) in \(\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots…
2016
The coefficient of \(x^{12}\) in \(\left(x^{3}+x^{4}+x^{5}+x^{6}+\dots \right)^{3}\) is ___________ .
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Correct answer: 10
Answer: 10
Method 1 — counting integer solutions: We seek the number of triples of integers a, b, c with a, b, c ≥ 3 and a + b + c = 12. Let y1 = a - 3, y2 = b - 3, y3 = c - 3, so y1, y2, y3 ≥ 0 and y1 + y2 + y3 = 3. The number of nonnegative integer solutions is "5 choose 2" = 10.
Method 2 — generating functions: Each factor is x^3 + x^4 + x^5 + … = x^3/(1 - x). Cubing gives x^9/(1 - x)^3. The coefficient of x^12 equals the coefficient of x^3 in (1 - x)^{-3}, which is the number of multisets of size 3 from 3 types: "5 choose 2" = 10.
Therefore the coefficient of x^12 is 10.