A group of three girls and three boys have tickets for six adjacent seats in one row of a theater. If the three boys will not sit in three adjacent seats, how many possible different seating arrangements are there for these 6 theatre-goers?	Answers.
A) 6! – 2!3!2!  B) 6! – 6!3!  C) 7 × 2!3!2!  D) 6! – 4!3!  E) 2!3!2!
There are 3 boys and 3 girls , we want to calculate the seating arrangements if three boys do not sit together, like MMM.   Let's calculate the # of arrangements when they SIT together and subtract from total # of arrangements of these 6 persons without restriction. Thus we'll get the # of arrangements asked in the question.   1. Total # of arrangements of 6 is 6!.   2. # of arrangements when 3 boys are seated together, like MMM;   Among themselves these 3 boys can sit in 3! # of ways,   Now consider these 3 boys as one unit like this {MMM}. We'll have total of 4 units: {MMM}{W}{W}{W} The # of arrangements of these 4 units is 4!.   Hence total # of arrangements when 3 men sit together is: 3!4!.   # of arrangements when 3 boys do not sit together would be: 6!-3!4!.   Answer: 6! – 4!3!
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A group of three girls and three boys have tickets for six adjacent seats in one row of a theater. If the three boys will not sit in three adjacent seats, how many possible different seating arrangements are there for these 6 theatre-goers?