Given the answer choices, I suspect that the prompt is intended to ask the following:
In how many ways can 6 people A, B, C, D, E and F be seated at a round table if A, B and C cannot sit in 3 adjacent seats such that A is between B and C?
a) 720
b) 120
c) 108
d) 84
e) 48
As noted in my post directly above, to count circular arrangements:
1. Place someone in the circle.
2. Count the number of ways to arrange the remaining people.
Good arrangements = (total possible arrangements) - (bad arrangements).
Total possible arrangements:
After A has been seated at the table, the number of ways to arrange the remaining 5 people = 5! = 120.
Bad arrangements:
In a bad arrangement, A, B and C sit in 3 adjacent seats, with A between B and C.
After A has been seated:
Number of options for the seat to the left of A = 2. (B or C.)
Number of options for the seat to the right of A = 1. (Must be B or C, whoever is not seated to the left of A.)
Number of ways to arrange the remaining 3 people = 3! = 6.
To combine these options, we multiply:
2*1*6 = 12.
Thus:
Good arrangements = 120 - 12 = 108.
The correct answer is
C.
The problem as written seems poorly worded.
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