Difference between revisions of "2005 Alabama ARML TST Problems/Problem 12"
(New page: ==Problem== Find the number of ordered pairs of positive integers <math>(a,b,c,d)</math> that satisfy the following equation:<center><math>a+b+c+d=12</math>.</center> ==Solution== The [[G...) |
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==Solution== | ==Solution== | ||
− | The [[ | + | The [[generating function]] for a is <math>1+x+x^2+x^3+\cdots</math>. The same for b, c, and d. |
<math>(1+x+x^2+x^3+\cdots)^4=1+4x+10x^2+\cdots</math> | <math>(1+x+x^2+x^3+\cdots)^4=1+4x+10x^2+\cdots</math> |
Revision as of 20:16, 5 January 2008
Problem
Find the number of ordered pairs of positive integers that satisfy the following equation:
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 .