Question

A railway carriage has 4 seats on each side. In how many ways can 6 men and 2 women sit if
(a) the women sit opposite each other
(b) the women do not sit next to each other?

Answer 1

(a)

Women sit opposite to each other means one woman at each side.

Let us start from left side 4 seats. Number of ways of selecting 1 woman for left side seats is C(2,1)= 2

Number of ways of selecting 3 men for left side seats is C(6,3) = 20

Now 3 men can arrange themselves in 3! on left side seats and woman can take any one of the four position. So possible number of ways of arranging left side 4 seats is

20 *2 *3! * 4 = 960

Now at right side woman can sit opposite to other woman in one way and man can arrange themselves in 3! ways so possible number of arrangements in right side is 3! = 6

So total number of ways 6 men and 2 women can sit if the women sit opposite each other is

960 *6 = 5760

Answer: 5760

(b)

Total number of people is: 6+2 = 8

Number of ways of these 8 people can arrange themsleves is 8! = 40320

Now suppose two men and two women ar at left side seats. So number of ways of selecting 2 men out of 6 is C(6,2) = 15

Now two women can sit together in 3 ways on left sides and women can arrange themselves in 2! ways and men in 2! ways. And men at right sides can arrange themselves in 4! ways. That is possible number of ways woman can sit together at left side is 3*2! *15 * 4! *2! = 4320

Likewise women can sit together at right side in 4320 ways. So possible number of ways women can sit together is: 4320 +4320 = 8640

So number of ways 6 men and 2 women can sit if the women do not sit next to each other is:

40320 - 8640 = 31680