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

Question

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

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

Bunuel · Accepted Answer

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. Similar questions to practice: seven-men-and-seven-women-have-to-sit-around-a-circular-92402.html a-group-of-four-women-and-three-men-have-tickets-for-seven-a-88604.html the-number-of-ways-in-which-5-men-and-6-women-can-be-seated-94915.html in-how-many-different-ways-can-4-ladies-and-4-gentlemen-be-102187.html 4-couples-are-seating-at-a-round-tables-how-many-ways-can-131048.html at-a-party-5-people-are-to-be-seated-around-a-circular-104101.html seven-family-members-are-seated-around-their-circular-dinner-102184.html seven-men-and-seven-women-have-to-sit-around-a-circular-11473.html a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html seven-men-and-five-women-have-to-sit-around-a-circular-table-98185.html a-group-of-8-friends-sit-together-in-a-circle-alice-betty-106928.html find-the-number-of-ways-in-which-four-men-two-women-and-a-106919.html Hope it helps.

Bunuel · Answer

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

GDR29 · Answer

Hi Bunuel, I find the same problem but with a rectangular table. How should be the reasoning there?

Thks

GDR29 · Answer

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?

plalud · Answer

Another way of finding the # of cases where the restriction is violated, i.e. when the two chief of staffs are together. # of cases when the two generals are together.

there are 7 spots. The first general who wants to be next to the other one. He can choose whatever spot (7). The second general that wants to be next to the first has 2 choices (2). The rest of them have their regular amount of choice, given that two spots are now occupied (5!).

Answer = 7*2*5!. If we care about relative positioning rather than absolute positioning, we divide that by 7 and found aforementionned answer.

For n slots, answer = n*2*(n-2)!

akadiyan · Answer

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

Number of ways 7 person can be arranged in a circular table = (N-1)! = (7-1)! = 6! = 720

Lets consider 2 people not sitting together as one entity = so we have 6 people and number of arrangements is 5!. The 2 people can be arranged in 2 ways so probability is 2*5! = 240

Total probability is = 720-240 = 480

Ans:   B

ScottTargetTestPrep · Answer

Recall that the circular permutations formula for n items is (n - 1)!

The number of ways for the Chief of Naval Operations (N) to sit next to Chief of the National Guard Bureau (G) is to treat the pair   as a single entity. The seating arrangement can be shown as:

- A - B - C - D - E

Since we have a circle, that above arrangement can be arranged in (6 -1)! = 5! = 120 ways. Additionally, we could seat N and G as  , giving us an additional (6 -1)! = 120 ways, for a total of 120 + 120 = 240 ways in which the two chiefs would sit next to each other.
The total number of ways for all seating arrangements is (7 - 1)! = 6! = 720.

Thus, the number ways the Chief of Naval Operations does not sit next to the Chief of the National Guard Bureau is 720 - 240 = 480.

Answer: B

rajeshforyouu · Answer

We have  5 Chiefs + Chief of Naval + Chief of NGB

Let's first arrange these  5 Chiefs  around a Circular Table:  4!  ways

Now we have 5 gaps among 5 Chiefs.
Out of 5 gaps we can select any 2:  5C2  ways
Chief of Naval & NGB can be arranged in  2!  ways

Answer: 4!*5C2*2! =   480

CEdward · Answer

Circular arrangements with restrictions:

(7-1)! = 720 <--- # of arrangements without restrictions

5! = 120 <--- # of instances where one chief sits on the left/right of the other

5! x 2 = 240 <--- sum of # of instances where one chief sits flanking either side

720 - 240 = 480.

Anser is B.

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