Bunuel wrote:
Tom, Bill, Robert, Roger, and Terry are standing in a row for a group photo. In how many different orders can the five men stand if Tom refuses to stand next to Roger?
A. 48
B. 64
C. 72
D. 96
E. 120
Here's another approach:
Take the task of arranging the 5 men and break it into
stages.
Stage 1: Arrange Bill, Robert, and Terry in a row
There are 3 people, so we can arrange them in
3! ways.
Now that we've arranged 3 men, we'll place a potential standing space on each side of these 3 men.
For example:
___ Terry ___ Robert ___ Bill ___ Notice that, when we place the 2 remaining men (Tom and Roger), in the 4 available spaces, we will be guaranteed that they are not next to each other.
Stage 2: Select a place for Tom to stand
There are 4 spaces available,, so we can complete this stage in
4 ways.
Stage 3: Select a place for Roger to stand
There are 3 remaining spaces, so we can complete this stage in
3 ways.
By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all 5 men) in
(3!)(4)(3) ways (= 72 ways)
Answer: C
Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this video from our course:
You can also watch a demonstration of the FCP in action here: