Seating arrangements of 7 boys and 5 girls in a row.

Question

In how many ways can these boys and girls be arranged in a row if between two particular boys A and B there are no boys but exactly 3 girls?

Answer 1

one way to approach this is to isolate the Middle Objective first, namely  A _ _ _ B or B _ _ _ A, where any 3 girls could be placed in the _ _ _ positions.  This would give us ₅P₃ =60 ways for the _ _ _ girls, times two for the AB or BA order of particular boys bracketing them.  So far, then this is 120 ways to compose (arrange) what I have called the Middle Objective.

 

Now we have to distribute the remaining people (5 other boys, 2 other girls, total of 7 persons).

 

We can treat the Middle Objective as a single entity, e.g. it could be at the beginning of the row, followed by a permutation of the 7 other persons, or one person could be placed at the beginning, followed by the Middle Objective single entity, followed by 6 other persons, etc.

 

This overall problem therefore can be solved by considering that we have 8 distinct things arrangeable in a row, so the overall final answer is ₈P₈ or 8!, times the factor of 120 that was developed for the internal structure of the Middle Objective.

 

120(8!).