answer = total permutations – no girls together
total permutations = (10-1)! = 9!
no girls together =
- select 5 boys and arrange them in alternate circular spaces = 6C5*(4!) = 6*24 = 144
- 5 spaces left choose one space for the one boy = 5C1 = 5
- remaining 4 places permute the 4 girls in circular order = (4-1)! = 6
- multiply them (since each will happen for each) = 144*5*6 = 4320
answer = 9! – 4320 = 358560