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answer = total permutations – no girls together 

 

total permutations = (10-1)! =  9!

 

no girls together =

  1. select 5 boys and arrange them in alternate circular spaces = 6C5*(4!) = 6*24 = 144
  2. 5 spaces left choose one space for the one boy = 5C1 = 5 
  3. remaining 4 places permute the 4 girls in circular order = (4-1)! = 6 
  4. multiply them (since each will happen for each) = 144*5*6 = 4320

  

answer = 9! – 4320 = 358560

 

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