Find the number of ways in which two Americans, two British, one Chinese, one Dutuch, and one Egyptian can sit on a round table so that persons of the same nationality are separated.

Explanation

The total number of persons without any restrictions is
`n(U)=(8-1)!`
`=7!=5040`
When, three americans `(A_(1),A_(2),A_(3))` are sit together,
`n(A)=5!xx3!`
`=720`
When, two british `(B_(1),B_(2))` are sit together
`n(B)=6!xx2!`
`=1440`
When, three americans `(A_(1),A_(2),A_(3))` and two british `(B_(1),B_(2))` are sit together `n(A capB)=4!xx3!xx2!=288`
`thereforen(AcupB)=n(A)+n(B)-n(A capB)`
`=720+1440-288=1872`
Hence, `n(AcapB')=n(U)-n(A cupB)`
`=5040-1872`
`=3168`