Question

for problems 3–4, an empty row at the stadium has nine seats, and all nine seats must be filled. there are 5 fans of the home team and 4 fans of the away team.

1. in the row, all 9 fans must be seated such that home and away fans are alternating. how many ways are there to arrange the 9 fans in the seats?

Explanation:

Step1: Check valid alternating patterns

With 9 seats, alternating pattern must start with home fan (since 5 home > 4 away): H, A, H, A, ..., H.

Step2: Calculate home fan arrangements

Arrange 5 distinct home fans: $5! = 120$.

Step3: Calculate away fan arrangements

Arrange 4 distinct away fans: $4! = 24$.

Step4: Total arrangements

Multiply both results: $120 \times 24 = 2880$.

Answer:

2880