Ways to Seat People Together in a Row

This problem asks for the number of ways 3 specific people can be seated in a row of 8 seats, with the condition that they must sit next to each other.

Understanding Adjacent Seating Arrangement

The core requirement is that the 3 people must sit in consecutive seats. We can simplify this by treating the group of 3 people as a single unit or block.

Step 1: Grouping the People

Consider the 3 people as one combined entity. Let's represent this block as [PPP].

Now, we have this block [PPP] and the remaining \( 8 - 3 = 5 \) empty seats. Effectively, we need to arrange this block and the 5 empty seats.

This means we have \( 1 \text{ block} + 5 \text{ empty seats} = 6 \) items to arrange in the row.

Step 2: Positioning the Group Block

We need to find how many places this block of 3 people can fit within the 8 seats. The block occupies 3 consecutive seats.

The possible positions for the block are:

• Seats 1, 2, 3
• Seats 2, 3, 4
• Seats 3, 4, 5
• Seats 4, 5, 6
• Seats 5, 6, 7
• Seats 6, 7, 8

Mathematically, the number of positions for a block of size \( k \) in a row of size \( n \) is \( n - k + 1 \). In this case, \( n = 8 \) and \( k = 3 \), so the number of positions is \( 8 - 3 + 1 = 6 \).

Step 3: Arranging Individuals Within the Group

Within the block of 3 seats occupied by the group, the 3 people can switch places among themselves. The number of ways to arrange 3 distinct people in 3 seats is given by the permutation of 3 items, which is \( 3! \).

Calculating the internal arrangements:

\( 3! = 3 \times 2 \times 1 = 6 \)

This means the 3 people can arrange themselves in 6 different orders within their block of seats.

Step 4: Total Arrangement Calculation

To find the total number of ways the 3 people can sit next to each other, we multiply the number of possible positions for the block by the number of ways the people can arrange themselves within the block.

Total Ways = (Number of positions for the block) \( \times \) (Number of arrangements within the block)

Total Ways = \( (8 - 3 + 1) \times 3! \)

Total Ways = \( 6 \times 6 \)

Total Ways = \( 36 \)

Final Answer Explanation

Thus, there are 36 distinct ways for 3 people to sit next to each other in a row of 8 seats. This corresponds to the value calculated.