Suppose that 10 women including Alice and Betty are arranged

Suppose that 10 women, including Alice and Betty, are arranged at random in a line. For each value of k between k=0 and k=8, find the probablitly that exactly k women are between Alice and Betty. I need the probability not the number of arrangements. (unless that actually is the probability)

Solution

There is 10 Women, along Alice, Betty
a)
When k=0, no women b/w Alice and Betty
Consider both as single entry and the total
arrangment in 2*(8+1)! = 2 *9! = 72576 ways

b)
When k=8, there is 8 b/w Alice and Betty
i.e ( A - - - - - - - - B)
and the total arrangment on 2*8! = 80640