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

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

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