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At a meeting of the 7 Joint Chiefs of Staff, the Chief of

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At a meeting of the 7 Joint Chiefs of Staff, the Chief of [#permalink] New post 11 Jun 2013, 03:02
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At a meeting of the 7 Joint Chiefs of Staff, the Chief of Naval Operations does not want to sit next to the Chief of the National Guard Bureau. How many ways can the 7 Chiefs of Staff be seated around a circular table?

A. 120
B. 480
C. 960
D. 2520
E. 5040
[Reveal] Spoiler: OA

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Re: At a meeting of the 7 Joint Chiefs of Staff, the Chief of [#permalink] New post 11 Jun 2013, 04:32
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emmak wrote:
At a meeting of the 7 Joint Chiefs of Staff, the Chief of Naval Operations does not want to sit next to the Chief of the National Guard Bureau. How many ways can the 7 Chiefs of Staff be seated around a circular table?

A. 120
B. 480
C. 960
D. 2520
E. 5040


7 people can be arranged around circular table in (7-1)!=6! # of ways.

Consider the two people (A and B) who do not want to sit together as one unit: {AB}. Now, 6 units {AB}, {C}, {D}, {E}, {F} and {G} can can be arranged around circular table in (6-1)!=5! # of ways. A and B can be arranged within their unit in two ways {AB} and {BA}. Thus the # of ways those two people sit together is 2*5!.

The number of ways they do not sit together = {Total} - {Restriction} = 6! - 2*5! = 5!(6 - 2) = 480.

Answer: B.

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Hope it helps.
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Re: At a meeting of the 7 Joint Chiefs of Staff, the Chief of [#permalink] New post 30 Mar 2014, 08:40
Hi Bunuel, I find the same problem but with a rectangular table. How should be the reasoning there?

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Re: At a meeting of the 7 Joint Chiefs of Staff, the Chief of [#permalink] New post 30 Mar 2014, 08:56
Bunuel, I'm also a little confused with the number of arrangements of n distinct objects in a circle. Why is it given by (n-1)!. In the veritas answer they say: "answer E (5040), should be the number of ways to arrange all 7 without the seating restriction given". Is this incorrect?
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Re: At a meeting of the 7 Joint Chiefs of Staff, the Chief of [#permalink] New post 30 Mar 2014, 10:29
Expert's post
GDR29 wrote:
Bunuel, I'm also a little confused with the number of arrangements of n distinct objects in a circle. Why is it given by (n-1)!. In the veritas answer they say: "answer E (5040), should be the number of ways to arrange all 7 without the seating restriction given". Is this incorrect?


I guess you did not follow any of the links given above...

The number of arrangements of n distinct objects in a row is given by n!.
The number of arrangements of n distinct objects in a circle is given by (n-1)!.

The difference between placement in a row and that in a circle is following: if we shift all object by one position, we will get different arrangement in a row but the same relative arrangement in a circle. So, for the number of circular arrangements of n objects we have:

\frac{n!}{n} = (n-1)!.

Theory on Combinations: math-combinatorics-87345.html

DS questions on Combinations: search.php?search_id=tag&tag_id=31
PS questions on Combinations: search.php?search_id=tag&tag_id=52

Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html

_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: At a meeting of the 7 Joint Chiefs of Staff, the Chief of   [#permalink] 30 Mar 2014, 10:29
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