Difference between revisions of "2012 AIME II Problems/Problem 3"

(Remove extra problem section)
(6 intermediate revisions by 5 users not shown)
Line 3: Line 3:
  
 
== Solution ==
 
== Solution ==
 
{{AIME box|year=2012|n=II|num-b=2|num-a=4}}
 
  
 
There are two cases:
 
There are two cases:
Line 12: Line 10:
 
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.
 
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 <math>{{2}\choose{1}} * {{2}\choose{1}} = 4</math> ways to choose one man and one woman. Thus there are <math>4^3 = 64</math> total possibilities conforming to case 1.
+
For the first case, in each department there are <math>{{2}\choose{1}} \times {{2}\choose{1}} = 4</math> ways to choose one man and one woman. Thus there are <math>4^3 = 64</math> total possibilities conforming to case 1.
  
For the second case, there is only <math>{{2}\choose{2}} = 1</math> way to choose two professors of the same gender from a department, and again there are <math>4</math> ways to choose one man and one woman. Thus there are <math>1 * 1 * 4 = 4</math> 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 <math>3! = 6</math> different department orders, so the total number of possibilities conforming to case 2 is <math>4 * 6 = 24</math>.
+
For the second case, there is only <math>{{2}\choose{2}} = 1</math> way to choose two professors of the same gender from a department, and again there are <math>4</math> ways to choose one man and one woman. Thus there are <math>1 \cdot 1 \cdot 4 = 4</math> 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 <math>3! = 6</math> different department orders, so the total number of possibilities conforming to case 2 is <math>4 \cdot 6 = 24</math>.
  
 
Summing these two values yields the final answer: <math>64 + 24 = \boxed{088}</math>.
 
Summing these two values yields the final answer: <math>64 + 24 = \boxed{088}</math>.
  
== See also ==
+
== See Also ==
 +
{{AIME box|year=2012|n=II|num-b=2|num-a=4}}
 +
{{MAA Notice}}
 +
 
 +
[[Category:Intermediate Combinatorics Problems]]

Revision as of 17:00, 9 August 2018

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 ${{2}\choose{1}} \times {{2}\choose{1}} = 4$ ways to choose one man and one woman. Thus there are $4^3 = 64$ total possibilities conforming to case 1.

For the second case, there is only ${{2}\choose{2}} = 1$ way to choose two professors of the same gender from a department, and again there are $4$ ways to choose one man and one woman. Thus there are $1 \cdot 1 \cdot 4 = 4$ 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 $3! = 6$ different department orders, so the total number of possibilities conforming to case 2 is $4 \cdot 6 = 24$.

Summing these two values yields the final answer: $64 + 24 = \boxed{088}$.

See Also

2012 AIME II (ProblemsAnswer KeyResources)
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. AMC logo.png