circular arrangement

Question

how many ways are there to arrange 6 girls and 15 boys in a circle such that there are atleast two boys between two adjacent girls?

Answer 1

For every girl there are 2 boys one at left and other at right. so for 6 girls, 12 boys can be chosen in 15P12 ways.Now every girl and 2 boys are considered as group remaining 3 boys(15-12 boys) are also consider as groups.so we have total 9 group in total and this can be arrange in circular in (9-1)!=8!ways. Number of permutation satisfy the conditions equals 15P12*8!

Answer 2

The boys can arrange them in 14! ways

there are 8 sits between atleat every 2 boys

Girls can arrange ⁸C₃ ways

6 girls can arrange 6! ways

So, total 14! ⨉  ⁸C₃ ⨉ 6! ways

Comments

how it is ensuring that there are atleast two boys between two adjacent girlsz?.check again.