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)
    none of these
Correct answer is option 'C'. Can you explain this answer?

CA Foundation Question

2 Answers
Sagarika Shinde answered Jun 14, 2018
When people are seated at a circular table, where the first person sits is IRRELEVANT. 
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. 

Muskan Jain answered 2 weeks ago
Out of the 3 ladies, if 2 are to sit together they can be seated in 3P2 = 6 ways.
Now, in the seats adjacent to the ladies only 2 gents can be seated because only 2 ladies are supposed to sit together.
Out of 3 gents, if 2 are to sit together they can be seated in 3P2 = 6 ways.
Now, the remaining two people can be seated in the remaining two seats in 2P2 = 2 ways.
The number of ways in which the 3 ladies and 3 gents can be seated at a round table so that any 2 and only 2 of the ladies sit together are
= 3P2 × 3P2 × 2P2
= 6 × 6 × 2
= 72.

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