Question

In how many ways can $7$ persons form a ring? In how many ways can $7$ Englishmen and $7$ Americans sit down at a round table, no two Americans being together?

Answer 1

$⟹$ In a ring $n$ persons can be arranged in $(n-1)!$ ways.
$\therefore$ No. of ways of forming a ring by $7$ persons $=6!=720$
Denoting $E⟶$ Englishmen
$A⟶$ Americans
Keeping $E_1$ fixed, other ${}^{\prime }{E}^{\prime }$s can be arranged in $6!$ ways, and the rest ${}^{\prime }{A}^{\prime }$s can be arranged in $7!$ as there are $7$ gaps between $7$ $E^{\prime }s$
$\therefore$ Total no. of ways $=7!\times 6!=3628800$