When people are seated at a circular table, where the first person sits is IRRELEVANT.Manjareev wrote:3 ladies and 3 gents can be seated at a round table so that any two and only two of the ladies sit together. the number of ways is
a. 70
b. 27
c. 72
d. 48
We need to count only the number of ways to arrange the remaining people RELATIVE to the first person seated.
Let the 3 women be A, B and C.
Case 1: A and B in adjacent seats
Once A is seated, the number of options for B = 2. (To the right or left of A.).
This AB block must be surrounded by men, so that 3 women are not in adjacent seats.
Number of options for the seat on the OTHER SIDE of A = 3. (Any of the 3 men.)
Number of options for the seat on the OTHER SIDE of B = 2. (Any of the 2 remaining men.)
Number of ways to arrange the 2 remaining people = 2! = 2.
To combine these options, we multiply:
2*3*2*2 = 24.
Remaining cases:
Since the same reasoning will apply to A and C in adjacent seats and to B and C in adjacent seats -- yielding 3 options for the two women in adjacent seats -- the result above must be multiplied by 3:
3*24 = 72.
The correct answer is C.
