Problem
COM-B2-M09-P010 Coefficient of a Rational Function
#10
★★★☆☆ Level 3 of 5
Find the coefficient of \(x^{10}\) in \(\frac{1}{(1-x)^2(1-x^3)}\).
Interpret the coefficient as the number of solutions to \(a+b+3c=10\).
The expansion \(\frac{1}{(1-x)^2(1-x^3)}=(1+x+x^2+\cdots)^2(1+x^3+x^6+\cdots)\) shows that the coefficient equals the number of triples \((a,b,c)\) such that \(a+b+3c=10\). For \(c=0,1,2,3\), the numbers of solutions for \(a+b\) are \(11,8,5,2\), respectively. The sum is \(11+8+5+2=26\).
Useful for moving from a formal series to a combinatorial interpretation.