mehaksal wrote:In how many different orders can the people Alice, Benjamin, Charlene, David, Elaine, Frederick, Gale, and Harold be standing on line if each of Alice, Benjamin, Charlene must be on the line before each of Frederick, Gale, and Harold?
1,008
1,296
1,512
2,016
2,268
Label the spaces #1, #2, .... #7, #8
Take the task of arranging all 8 people and break it into stages.
Stage 1: Place David
There are 8 spaces available, so this stage can be accomplished in
8 ways.
Stage 2: Place Elaine
There are 7 spaces remaining, so this stage can be accomplished in
7 ways.
Important: At this point, there are 6 spaces remaining, and Alice, Benjamin, Charlene must occupy the 3 spaces that are ahead of the last 3 spaces. We'll call these the "front spaces"
Stage 3: Place Alice
Alice must occupy one of the 3 front spaces. So this stage can be accomplished in
3 ways.
Stage 4: Place Benjamin
Benjamin must occupy one of the 2 remaining front spaces. So this stage can be accomplished in
2 ways.
Stage 5: Place Charlene
Charlene must occupy the last remaining front space. So this stage can be accomplished in
1 way.
At this point, there are 3 spaces remaining.
Stage 6: Place Frederick
There are 3 spaces remaining. So this stage can be accomplished in
3 ways.
Stage 7: Place Gale
There are 2 spaces remaining. So this stage can be accomplished in
2 ways.
Stage 8: Place Harold
There is 1 space remaining. So this stage can be accomplished in
1 way.
By the Fundamental Counting Principle (FCP) we can complete all 8 stages (and thus arrange all 8 people) in
(8)(7)(3)(2)(1)(3)(2)(1) ways ([spoiler]= 2016 ways[/spoiler])
Answer =
D
Cheers,
Brent
Aside: For more information about the FCP, we have a free video on the subject:
https://www.gmatprepnow.com/module/gmat-counting?id=775
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