GreenlightTestPrep wrote:
Carcass wrote:
Team A and Team B are competing against each other in a game of tug-of-war. Team A, consisting of 3 males and 3 females, decides to lineup male, female, male, female, male, female. The lineup that Team A chooses will be one of how many different possible lineups?
(A) 9
(B) 12
(C) 15
(D) 36
(E) 720
Take the task of lining up the 6 competitors and break it into
stages.
Stage 1: Select a competitor for the 1st position
This person must be a male.
Since there are 3 males to choose from, we can complete stage 1 in
3 ways
Stage 2: Select a competitor for the 2nd position
This person must be a female.
Since there are 3 females to choose from, we can complete stage 2 in
3 ways
Stage 3: Select a competitor for the 3rd position
This person must be a male.
There are 2 males remaining to choose from (since we already selected a male in stage 1), so we can complete stage 3 in
2 ways
Stage 4: Select a competitor for the 4th position
This person must be a female.
There are 2 females remaining to choose from. So we can complete stage 4 in
2 ways
Stage 5: Select a male for the 5th position
There's only 1 male remaining. So we can complete stage 5 in
1 way
Stage 6: Select a female for the 6th position
There's only 1 female remaining. So we can complete stage 6 in
1 way
By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus create a 6-person lineup) in
(3)(3)(2)(2)(1)(1) ways (= 36 ways)
Answer: D
Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.
Adding to this answer is the question:
When can we use the Fundamental Principle of Counting?
When? When the number of outcomes for each stage is independent of the other stages. In this case it is. That is for example, no matter what male we choose in stage 1, there will always be 3 outcomes to choose from in stage 2, 2 outcomes to choose from in stage 3, and so on.
For example in what case would you not be able to use Fundamental Principle of Counting?
Let's say
Males: Mike, Marco, and Mills
Females: Jess, Jill, Gina
Suppose we had the requirement that if we choose Mike in Stage 1 we can only choose Jess and Jill in stage 2, and if we choose Marco or Mills in stage 1 we can choose any of the girls in stage 2.
Then here we wouldn't be able to use Fundamental Principle of Counting, because the number of outcomes of the stages are not independent of each other. If Mike is chosen we only have 2 possible outcomes in Stage 2, and if we choose Marco or Mills we have 3 possible outcomes in Stage 2.