There are n people seated randomly in n seats. The question asks how many of the n! possible arrangements have Ms. Jones and Ms. Smith sitting next to each other. The solution is (n-1)!*2!, as Jones and Smith can sit in 2 arrangements, and the remaining n-1 people can be arranged in (n-1)! ways, forming the total number of arrangements where Jones and Smith sit together.

1. Suppose n people are seated at random in a row of n seats. How many of the n! possible
arrangements of people have Ms. Jones and Ms. Smith sitting next to each other?
Solution
(n-1)!*2!
Jones and smith can sit in two ways (Jones , Smith ) or (Smith , Jones). This forms one pair
Hence we have n-1 people remaining which arrrange them selves in (n-1)! ways.