A school principal, his wife, and three other teachers are to be seated in a row. The principal and his wife must sit next to each other. Find the number of ways this can be done.

1. Problem Description

A school principal, his wife, and three other teachers are to be seated in a row. The principal and his wife must sit next to each other. Find the number of ways this can be done.

2. Solution Steps

There are a total of 1 (principal) + 1 (wife) + 3 (teachers) = 5 people to be seated.

Since the principal and his wife must sit next to each other, we can treat them as a single unit. This means we effectively have 4 entities to arrange: (principal, wife) and the 3 teachers.

The number of ways to arrange these 4 entities is $4!$.

$4! = 4 \times 3 \times 2 \times 1 = 24$.

However, the principal and his wife can switch places within their unit. The number of ways to arrange the principal and his wife within their unit is $2! = 2 \times 1 = 2$.

Therefore, the total number of ways to seat the 5 people such that the principal and his wife sit next to each other is the product of the arrangements of the 4 entities and the arrangements of the principal and wife within their unit.

Total number of arrangements = $4! \times 2! = 24 \times 2 = 48$.

3. Final Answer

The number of ways this can be done is 48.