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# M01, Q24

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Intern
Joined: 28 Dec 2011
Posts: 9
Location: United States (IL)
GMAT 1: 750 Q48 V44
GPA: 2.84
WE: Military Officer (Military & Defense)
Followers: 0

Kudos [?]: 6 [0], given: 1

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29 Dec 2011, 15:55
If 5 noble knights are to be seated at a round table, then how many different ways can they be seated?

120
96
60
35
24

Fix a knight in a spot to arrange the rest. The formula is (5-1)!, or 4!

Why is the answer 4! and not (5!)?

I can't wrap my head around that.

When I see 5 people, 5 empty spots, I automatically think - ok

5x4x3x2x1 = 5!

1st seat can have any of the 5
2nd seat will have one of the remaining 4...etc etc etc.

_________________

"Never doubt that a small group of thoughtful, committed citizens can change the world. Indeed, it's the only thing that ever has" - Margaret Mead

Magoosh GMAT Instructor
Joined: 28 Dec 2011
Posts: 3305
Followers: 1122

Kudos [?]: 4925 [2] , given: 54

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29 Dec 2011, 16:56
2
KUDOS
Expert's post
Great question! I'm happy to help!

You are SO close --- in fact, if we were seating these five knights in a straight line (for example, on a long bench), then the number of arrangements would be 5! = 120

Suppose the 5 knights are A, B, C, D, and E.

If they sit as ABCDE vs.CDEAB on a bench, those are two very different arrangements. At a round table though (assuming there are five seats equally spaced around the table), those two are identical arrangements: each knight has the same two neighbors on his same respective sides. These two seating plans are shown in the diagram below.

So, really the question is: let's say A sits in a seat --- that could be any seat, since they are all symmetrical. Once A has sat, there are 4! ways for the other four knights to fill those other four spaces. Thus, only 4! possibilities.

I hope that was clear and helpful. Please let me know if you have any questions.

Mike
Attachments

knights' seating plans.PNG [ 25.5 KiB | Viewed 833 times ]

_________________

Mike McGarry
Magoosh Test Prep

Intern
Joined: 28 Dec 2011
Posts: 9
Location: United States (IL)
GMAT 1: 750 Q48 V44
GPA: 2.84
WE: Military Officer (Military & Defense)
Followers: 0

Kudos [?]: 6 [0], given: 1

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29 Dec 2011, 20:19
Once I posted, the similar threads thing on the bottom popped up and I found the answer. But your explanation was very clear and concise. Thank you.
_________________

"Never doubt that a small group of thoughtful, committed citizens can change the world. Indeed, it's the only thing that ever has" - Margaret Mead

Re: M01, Q24   [#permalink] 29 Dec 2011, 20:19
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# M01, Q24

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