I am facing some issues. My approach was the following -

Carla is hosting a dinner party with 5 guests and herself, making a total of 6 people. She will sit at the head of the table, leaving 5 seats for her guests. The constraint is that no two guests of the same gender can sit next to each other. This means the seating arrangement must alternate genders.

There are two possible gender arrangements for the guests around the table:

1. Boy, Girl, Boy, Carla, Boy, Girl (BGBCBG)
2. Girl, Boy, Carla, Boy, Girl, Boy (GBCBGB)

For each of these arrangements, we can calculate the number of ways the guests can be seated based on their gender.

Assume Carla is fixed

**For arrangement 1 (BGBCBG):**

- There are 3 boys and 2 girls.
- For the first slot for a boy, there are 3 choices (3 boys).
- For the first slot for a girl, there are 2 choices (2 girls).
- For the second slot for a boy, there are 2 choices (the remaining 2 boys).
- For the second slot for a girl, there is 1 choice (the remaining girl).
- For the third slot for a boy, there is 1 choice (the remaining boy).

This gives us \(3 \times 2 \times 2 \times 1 \times 1 = 12\) arrangements for the first scenario.

**For arrangement 2 (GBCBGB):**

- For the first slot for a girl, there are 2 choices (2 girls).
- For the first slot for a boy, there are 3 choices (3 boys).
- For the second slot for a girl, there is 1 choice (the remaining girl).
- For the second slot for a boy, there are 2 choices (the remaining 2 boys).
- For the third slot for a boy, there is 1 choice (the remaining boy).

This would also give us \(2 \times 3 \times 1 \times 2 \times 1 = 12\) arrangements for the second scenario.

i.e. a total = 24

One of these is the reverse of the other one, but since order matters. Won't both of these orders be correct?

I know I'm wrong, but what's the fundamental flaw in my reasoning? I'm not able to understand. Can you please help me figure that part out?