Bunuel wrote:
An office comprised of eight employees is planning to have a foosball game. A matchup consists of four players, split into pairs. If any employee can be paired up with any other employee, then how many unique matchups result?
(A) 70
(B) 210
(C) 280
(D) 336
(E) 420
Take the task of creating a matchup and break it into
stages.
Stage 1: Select 4 employees
Since the order in which we select the employees does not matter, we can use combinations.
We can select 4 employees from 8 employees in 8C5 ways (70 ways)
So, we can complete stage 1 in
70 ways
Stage 2: Divide the 4 selected employees into 2 teams
Let's say the 4 selected employees are A, B, C, D
A nice way to determine the number of ways to divide the 4 employees into 2 teams is to find a partner for one person.
For example, let's find a partner for employee A.
NOTE: once we choose a partner for employee A then, by default, the remaining two two employees will be paired together.
In how many ways can we select a partner for employee A? Well, A can be paired with B, C or D
So, we can complete stage 2 in
3 ways
ASIDE: The 3 pairings are:
AB vs CD
AC vs BD
AD vs BC
By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus create a matchup) in
(70)(3) ways (= 210 ways)
Answer:
Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.
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