Question

A round table conference is to be held between delegates of $20$ countries. In how many ways can they be seated if two particular delegates are:
a) Always together?
b) Never together?

Answer 1

$\left(a\right)$ Let the two particular deligate who wish to rit together be reated as one deligate So we have $19$ delegate who can be arange an a round table in $(19-1)!=18!$ ways. The particular delegate can be seated in $2$ ways So Total ways $=2\times 18!$

$\left(b\right)$ The total number of arrangement of $20$ delegate an a round table is $19!$

So the number of arrangement of $20$ delegates on a round table is $19!-2\times 18!$

$=17\times 18!$