A group of four Womens and three Mens have tickets for seven adjacent seats in one row of a theater. If the three mens will not sit in three adjacent seats, how many possible different seating arrangements are there for these 7 theatre-goers?
Answers.
A) 7! – 5!3!
B) 7! – 4!3!
C) 7 × 2!3!2!
D) 7! – 2!3!
E) 2!3!2!
There are 3 boys and 4 girls , we want to calculate the seating arrangements if three mens 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 mens 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 mens do not sit together would be: 7!-3!5!.   7! – 5!3!nswer: 7! – 5!3!).
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A group of four Womens and three Mens have tickets for seven adjacent seats in one row of a theater. If the three mens will not sit in three adjacent seats, how many possible different seating arrangements are there for these 7 theatre-goers?