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The circular table question (permutations & combinations

Problem Solving — algebra and arithmetic (GMAT Focus Edition)
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The circular table question (permutations & combinations

by anujan007 » Thu Jul 05, 2012 4:13 pm
If 6 people are going to sitting at a round table, but Sam will not sit next to Suzie, how many different ways can the group of 6 sit?
A) 120
B) 108
C) 96
D) 74
E) 56

OA : C
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by anujan007 » Thu Jul 05, 2012 4:21 pm
I encountered this question on one of the sites with sample questions. I was absolutely stumped by this question and want to understand the way to approach such problems. I do not know the difficulty level of this question if at all it appears on the actual GMAT and also whether this is a standard question. Nevertheless it will surely not hurt to know how to approach such problems.

My approach was :

Sam can choose a place in 6C1 ways i.e. 6
Suzie can now choose a place 3! ways from the remaining 2 seats i.e. 6 ways
The 2 seats next to Sam and the 2 seats not taken by Suzie can again be taken 4! ways i.e. 24

I for sure knew that I was wrong since the answer of 36+24=60 was nowhere in the answer choices. Even to guess from here on I had no idea of what to look out for. My answer was finally A which was wrong.
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by stewari » Fri Jul 06, 2012 12:29 am
Total no of combinations = (n-1)! = 5!

So no of combinations where Sam and Suzie dont sit together will be 5!-(2*4!)
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by theCEO » Fri Jul 06, 2012 12:34 am
total ways to sit = 5!
A B C D E F
1X5X4X3X2X1

ways for them to sit together = 4!x2
(A B C D E F)
2*(1X1X4X3X2X1)

ways for them to sit apart = 5! - 4!2 = 72
Last edited by theCEO on Fri Jul 06, 2012 3:24 am, edited 1 time in total.
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by stewari » Fri Jul 06, 2012 12:41 am
Hmm...Sam and Suzie can swap places for each 4! arrangement right? So it should be 2*4!. Correct me if I am wrong?
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by eagleeye » Fri Jul 06, 2012 2:02 am
anujan007 wrote:If 6 people are going to sitting at a round table, but Sam will not sit next to Suzie, how many different ways can the group of 6 sit?

OA : C
Couple of of ways of doing this:

First:
a. Total circular permutations = (6-1)! = 5! = 120.
b. Ways in which Sam and Suzie sit together = 2! * 4! = 2*24 = 48
Required ways = Total - Together = 120 - 48 = 72.

Second:
a. We have total of 6 places. Fix Suzie. Now Sam can't sit at either seat beside her. So number of places where Sam can sit = 5-2 = 3.
For the other 4 people we can arrange them in 4! ways in 4 seats.
So total ways = 3 * 4! = 72.

You can refer to my posts here as well.
https://www.beatthegmat.com/round-table- ... tml#483062
https://www.beatthegmat.com/round-circul ... tml#483072

Let me know if this helps :)
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by theCEO » Fri Jul 06, 2012 3:25 am
stewari wrote:Hmm...Sam and Suzie can swap places for each 4! arrangement right? So it should be 2*4!. Correct me if I am wrong?
You are right :)
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by anujan007 » Sun Jul 08, 2012 12:46 pm
eagleeye wrote: Couple of of ways of doing this:

First:
a. Total circular permutations = (6-1)! = 5! = 120.
b. Ways in which Sam and Suzie sit together = 2! * 4! = 2*24 = 48
Required ways = Total - Together = 120 - 48 = 72.

Second:
a. We have total of 6 places. Fix Suzie. Now Sam can't sit at either seat beside her. So number of places where Sam can sit = 5-2 = 3.
For the other 4 people we can arrange them in 4! ways in 4 seats.
So total ways = 3 * 4! = 72.

You can refer to my posts here as well.
https://www.beatthegmat.com/round-table- ... tml#483062
https://www.beatthegmat.com/round-circul ... tml#483072

Let me know if this helps :)
Hey there eagleeye,

I browsed through your solution and though the reasoning looks sound, the answer does not match. The actual answer is C i.e. 96. You need to add another 4! ways for Sam not being next to Suzie.
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by eagleeye » Sun Jul 08, 2012 3:00 pm
anujan007 wrote:
Hey there eagleeye,

I browsed through your solution and though the reasoning looks sound, the answer does not match. The actual answer is C i.e. 96. You need to add another 4! ways for Sam not being next to Suzie.
Hi anujan007:

The reasoning and the final answer of 72 is correct. Wherever you got 96 from has the wrong answer posted. This is why it is important to stick to reputed sources such as the GMAC and other reputable test prep companies for questions. Some sites aren't too trustworthy.
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by Jim@StratusPrep » Sun Jul 08, 2012 4:00 pm
eagleeye wrote:
Second:
a. We have total of 6 places. Fix Suzie. Now Sam can't sit at either seat beside her. So number of places where Sam can sit = 5-2 = 3.
For the other 4 people we can arrange them in 4! ways in 4 seats.
So total ways = 3 * 4! = 72.
This is the most repeatable way to do the problem.
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