Answer :
Answer:
Step-by-step explanation:
Given
There are n people in a row
So there are n! ways to arrange n people in a row
When k people is between A and B such that
[tex]0\leq k\leq n-2[/tex]
as A and B sits outside of k person
there are [tex]2\times \left [ n-(k+2)+1\right ][/tex] ways to arrange A and B with K sits between them
[tex]=2\left [ n-k-1\right ][/tex]
For each of those seating arrangement, there are [tex](n-2)![/tex] ways to arrange n-2 people
Using Multiplication Principle there [tex]2\left [ n-k-1\right ]\times (n-2)![/tex] ways to arrange n people such that k people seats between A and B.
Therefore Probability that there are k people between A and B
[tex]=\frac{2\left [ n-k-1\right ]\times (n-2)!}{n!}[/tex]