Answer to Question #137804 in Combinatorics | Number Theory for Geetansh

Question

The number of ways in which 4 women and 10 men are to be seated in a row so that exactly 3 men sit between every 2 nearest women is P. Then the value of P/10! is equal to

Answer 1

There are 10 men and 4 women.

Between two nearest women there will be exactly 3 men .

So after grouping of 3 men , there will be 1 man extra who will have to seat at any one of two ends. That looks like follows

M W MMM W MMM W MMM W

or

W MMM W MMM W MMM W M

For first case , We arrange 10 men in 10! ways. After that 4 women can seat in 4 places [ where W is written in first case ] by 4! ways. So number of arrangments is (10!)x(4!)

Similarly for second case , number of arrangments is (10!)x(4!)

So total number of arrangments is 2x(10!)x(4!) .

Therefore P = 2x(10!)x(4!) .

=> P/10 = 2x(10!)x(4!) /10 = 2x(9!)x(4!)

=> P/10 = 17418240