A group of four girls and three boys have tickets for seven 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 7 theatre-goers? Choose the most appropriate option
A) 7! – 2!3!2! B) 7! – 4!3! C) 7 × 2!3!2! D) 7! – 5!3! E) 2!3!2!
There are 3 boys and 4 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 7 persons without restriction. Thus we'll get the # of arrangements asked in the question.   1. Total # of arrangements of 7 is 7!.   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 5 units: {MMM}{W}{W}{W}{W}. The # of arrangements of these 5 units is 5!.   Hence total # of arrangements when 3 men sit together is: 3!5!.   # of arrangements when 3 boys do not sit together would be: 7!-3!5!.   Answer: 7! – 5!3!).
<result>D</result>
A group of four girls and three boys have tickets for seven 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 7 theatre-goers?