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

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

Thks

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?

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)!

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?

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

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 [N - G] as a single entity. The seating arrangement can be shown as:

[N - G] - 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 [G - N], 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

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

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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