Difference between revisions of "2012 AIME II Problems/Problem 3"
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== Solution 2== | == Solution 2== | ||
| − | Use generating functions. For each department, there is 1 way to pick 2 males, 4 ways to pick one of each, and 1 way to pick 2 females. Since there are three departments in total, and we wish for three males and three females, the answer will be equal to the coefficient of <math>x^3y^3</math> in the expansion of <math>(x^2+4xy+y^2)^3</math>. The requested coefficient is <math>\boxed{088}</math> | + | Use [[generating functions]]. For each department, there is 1 way to pick 2 males, 4 ways to pick one of each, and 1 way to pick 2 females. Since there are three departments in total, and we wish for three males and three females, the answer will be equal to the coefficient of <math>x^3y^3</math> in the expansion of <math>(x^2+4xy+y^2)^3</math>. The requested coefficient is <math>\boxed{088}</math> |
~sigma | ~sigma | ||
Latest revision as of 00:26, 28 January 2023
Contents
Problem 3
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.
Solution
There are two cases:
Case 1: One man and one woman is chosen from each department.
Case 2: Two men are chosen from one department, two women are chosen from another department, and one man and one woman are chosen from the third department.
For the first case, in each department there are 
ways to choose one man and one woman. Thus there are 
total possibilities conforming to case 1.
For the second case, there is only 
way to choose two professors of the same gender from a department, and again there are 
ways to choose one man and one woman. Thus there are 
ways to choose two men from one department, two women from another department, and one man and one woman from the third department. However, there are 
different department orders, so the total number of possibilities conforming to case 2 is 
.
Summing these two values yields the final answer: 
.
Solution 2
Use generating functions. For each department, there is 1 way to pick 2 males, 4 ways to pick one of each, and 1 way to pick 2 females. Since there are three departments in total, and we wish for three males and three females, the answer will be equal to the coefficient of 
in the expansion of 
. The requested coefficient is 
~sigma
See Also
| 2012 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
