2005 Alabama ARML TST Problems/Problem 12
Problem
Find the number of ordered pairs of positive integers that satisfy the following equation:
![$a+b+c+d=12$](http://latex.artofproblemsolving.com/c/e/6/ce6136616ccd8c46acf5b3c5add0fcd7203037b3.png)
Solution
The generating function for a is . The same for b, c, and d.
Since the existance of the Binomial Theorem, we can assume that these are the results of choosing.
Checking , it works.
We want the coefficient of x^12, so we have .