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: Determine valid alternating pattern

Since there are 5 home fans (H) and 4 away fans (A), the only valid alternating pattern for 9 seats is H-A-H-A-H-A-H-A-H (starts with home, as starting with away would need equal or more away fans).

Step2: Calculate home fan arrangements

Home fans are distinct: number of ways = $5! = 120$.

Step3: Calculate away fan arrangements

Away fans are distinct: number of ways = $4! = 24$.

Step4: Total arrangements

Multiply home and away ways: $5! \times 4! = 120 \times 24$.

Answer:

2880