Question

Find the number of ways of arranging $6$ boys and $6$ girls arround a circular table so that

1. all the girls sit together
   
2. no two girls sit together
   
3. boys and girls sit alternately
   

Answer 1

$6$ boys and $6$ girls

No of ways all girls sit together $=5!\times 6!$

$5!$ for arranging girls and $6!$ for arranging boys

No. of ways no two girls sit together $=5!\times 6!\times {{}^{6}C}_6=5!\times 6!$

$5!$ for girls and $6!$ for boys

No. of ways boys and girls sit alternatively $=5!\times 6!\times {{}^{6}C}_6=5!\times 6!$

$5!$ for girls and $6!$ for boys