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counting techniques
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iamnikkilarin
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in the finals of 100 meter dash, eight runners compete for the 1st place, 2nd place and 3rd place. How many different ways can the runner finish 1st, 2nd and 3rd?
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Brent@GMATPrepNow
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I'm not 100% sure I follow the question.iamnikkilarin wrote:in the finals of 100 meter dash, eight runners compete for the 1st place, 2nd place and 3rd place. How many different ways can the runner finish 1st, 2nd and 3rd?
Here's how I read it.
How many different race results (with all 8 runners) are possible if one runner (say Fred) must finish 1st, 2nd or 3rd?
My approach.
Take the task of building race results and break it into stages.
Stage 1: Place Fred in the 1st, 2nd or 3rd position.
This stage can be accomplished in 3 ways.
Stage 2: Place a different runner in one of the 7 remaining positions.
This stage can be accomplished in 7 ways.
Stage 3: Place a different runner in one of the 6 remaining positions.
This stage can be accomplished in 6 ways.
Stage 4: Place a different runner in one of the 5 remaining positions.
This stage can be accomplished in 5 ways.
.
.
.
.
Stage 8: Place the last remaining runner in the last remaining position.
This stage can be accomplished in 1 way.
At this point, we can apply the Fundamental Counting Principle (FCP) and say that the total number of race results = 3x7x6x5x4x3x2x1
Cheers,
Brent
PS: For more information about the FCP, here's a free video: https://www.gmatprepnow.com/module/gmat-counting?id=775
Brent Hanneson - Creator of GMATPrepNow.com
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Kobe_Kassidy
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Hey Brent,Brent@GMATPrepNow wrote:I'm not 100% sure I follow the question.iamnikkilarin wrote:in the finals of 100 meter dash, eight runners compete for the 1st place, 2nd place and 3rd place. How many different ways can the runner finish 1st, 2nd and 3rd?
Here's how I read it.
How many different race results (with all 8 runners) are possible if one runner (say Fred) must finish 1st, 2nd or 3rd?
My approach.
Take the task of building race results and break it into stages.
Stage 1: Place Fred in the 1st, 2nd or 3rd position.
This stage can be accomplished in 3 ways.
Stage 2: Place a different runner in one of the 7 remaining positions.
This stage can be accomplished in 7 ways.
Stage 3: Place a different runner in one of the 6 remaining positions.
This stage can be accomplished in 6 ways.
Stage 4: Place a different runner in one of the 5 remaining positions.
This stage can be accomplished in 5 ways.
.
.
.
.
Stage 8: Place the last remaining runner in the last remaining position.
This stage can be accomplished in 1 way.
At this point, we can apply the Fundamental Counting Principle (FCP) and say that the total number of race results = 3x7x6x5x4x3x2x1
Cheers,
Brent
PS: For more information about the FCP, here's a free video: https://www.gmatprepnow.com/module/gmat-counting?id=775
I think the question is more basic than that. I think it's just asking how many different orderings of 1st, 2nd, and 3rd place are possible from the group of 8 runners.
To the original poster, if this is indeed the case, the answer is [spoiler]8*7*6 = 336[/spoiler]
