How Many Ways Can They Be Seated?
Eight people attend a meeting, sitting around a circular table. The four officers must sit in a row (next to one another), and the four other people may sit anywhere. How many ways can the eight people be seated?
Note: Two arrangements are considered distinct even if one is a rotation of the other.
The answer is 4608.
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We can designate 4 seats for the officers, because they must sit next to each other. The number of ways they can be seated in the 4 seats is 4! or 24 .
The 4 other people may sit in 4! or 24 ways in the 4 other seats.
Each order they can be seated in has 8 rotations.
We can multiply the three numbers we have gotten. 2 4 × 2 4 × 8 = 4 6 0 8