Question

The number of ways, 5 boys and 4 girls can sit in a row so that either all the boys sit together or no two boys sit together is:

Hint:

Use casework to handle complex seating arrangements. Consider treating groups as units for easier calculation.

Answer 1

Correct Answer: 17280

To determine the number of seating arrangements for 5 boys and 4 girls where either all boys sit consecutively or no two boys sit consecutively, we proceed as follows:

Scenario 1: All boys are seated together.

Consider the 5 boys as a single unit. This unit, along with the 4 girls, forms 5 distinct entities to be arranged.

The number of permutations for these 5 entities is 5!:

5! = 120

Within the group of 5 boys, their internal arrangement can be done in 5! ways:

5! = 120

 

The total arrangements for this scenario are:

120 × 120 = 14400

 

Scenario 2: No two boys are seated together.

First, arrange the 4 girls in 4! ways:

4! = 24

 

This creates 5 potential positions for the boys (before the first girl, between each pair of girls, and after the last girl). We must place the 5 boys in these 5 positions, which can be done in 5! ways:

5! = 120

 

The total arrangements for this scenario are:

24 × 120 = 2880

 

Total arrangements for both scenarios:

14400 + 2880 = 17280

 

The resulting value of 17280 is confirmed to be within the specified range.