Problem
COM-B2-M09-P006 Even Summands
#6
★★★☆☆ Level 3 of 5
How many quadruples of nonnegative even integers \((a,b,c,d)\) satisfy \(a+b+c+d=18\)?
Set \(a=2a'\), \(b=2b'\), \(c=2c'\), \(d=2d'\), or use the factor \(1+x^2+x^4+\cdots\).
The generating function is \(\frac{1}{(1-x^2)^4}\). We need the coefficient of \(x^{18}\), equivalently the coefficient of \(y^9\) in \(\frac{1}{(1-y)^4}\). This is \(\binom{9+3}{3}=\binom{12}{3}=220\).
Emphasizes that a change of variable often simplifies the coefficient.