Question

I'm going to cry.

Mary, three female friends, and her brother, Peter, attend the theatre. In the theatre there is a row of 10 empty seats. For the first half of the show, they decide to sit next to each other in this row.

Find the number of ways these five people can be seated in this row.

Please explain why the answer is 6x5! and not just 5!

Answer 1

Explanation:

To find the number of ways that Mary, her three female friends, and her brother Peter can be seated in a row of 10 seats, we can follow these steps:

First, we need to consider the number of ways that the five people can sit together in the row of 10 seats. We can treat these five people as a single unit and arrange them in the row in 5! (5 factorial) ways.

However, since Mary and her friends are all female, we need to account for the number of ways that they can be arranged within this group of five. We can arrange the four women in 4! (4 factorial) ways.

Finally, we need to consider the number of ways that Peter can be seated in one of the remaining seats. Since there are six remaining seats, Peter can be seated in any of these seats in 6 ways.

Therefore, the total number of ways that Mary, her three female friends, and her brother Peter can be seated in the row of 10 seats is:

5! × 4! × 6 = 120 × 24 × 6 = 17,280

So there are 17,280 ways that these five people can be seated in the row of 10 seats, assuming that they all sit next to each other for the first half of the show.