3 Couples in a Circle

Probability Level 4

3 married couples go out for dinner and are seated randomly around a circular table for six people. What is the probability that exactly and only 1 of the 3 couples are seated together?

You can solve 3 Couples in a Circle -- II here

The answer is 0.4.

1 solution

Satyen Nabar
Sep 24, 2014

Fix a couple seated together. They can change places (2 x). The other couples can only sit one seat apart. They can exchange places (2 x2x). they can exchange seats with the other couple (2x). Thus there are 2x2x2x2= 16 positions when one couple is fixed. Since fixing other two couples will lead to different seating, the total number of ways is 16x3 = 48.

6 people can sit in a circle in 5! ways = 120.

Probability of exactly one couple seated together = 48/120 = 0.4

but in the mentioned 48 ways there are exactly 4 cases in which the other people sit with their own couples .Isn't it,hence the answer should be:(48-4)/(5!)=0.36666

Kunal Gupta - 6 years, 8 months ago

Consider that the 3 couples are

1,2

3,4,

5,6

Consider six seats in a circle numbered from top clockwise.

Put 1 and 2 in seats one and two. They can exchange places (2x)

A) Now put the others in the clockwise order 3, 6 and 4, 5. They can exchange individual places but only simultaneously 6,3, 5, 4. (2x).

The couples can change places with each other so 4, 5, and 3, 6. (2x)

Total = 8 ways.

B) Now put the others in the clockwise order 3, 5 and 4, 6. Same situation as above.

Total = 8 ways.

Grand total of 16 ways keeping 1, 2 in first two seats.

The ways of selecting any 1 couple out of 3 is 3 C1 = 3 ways.

So 16 * 3 = 48.

I hope this makes it clear.

Satyen Nabar - 6 years, 8 months ago

3 Couples in a Circle

The answer is 0.4.

1 solution

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