Problem
COM-B2-M09-P002 Three Unrestricted Variables
#2
★★☆☆☆ Level 2 of 5
How many triples of nonnegative integers \((a,b,c)\) satisfy \(a+b+c=14\)? Solve the problem using a generating function.
Each variable gives the factor \(1+x+x^2+\cdots\).
The desired number is the coefficient of \(x^{14}\) in \(\frac{1}{(1-x)^3}\). Since \(\frac{1}{(1-x)^3}=\sum_{n\ge 0}\binom{n+2}{2}x^n\), the answer is \(\binom{16}{2}=120\).
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