1. **Problem statement:** We have 5 people (A, B, C, D, E) and 5 chairs in a row. We want to find the number of seating arrangements where A and B are **not** sitting next to each other. 2. **Total arrangements without restriction:** The total number of ways to seat 5 people in 5 chairs is $$5! = 120$$. 3. **Calculate arrangements where A and B sit together:** Treat A and B as a single block. This block plus the other 3 people (C, D, E) makes 4 entities to arrange. Number of ways to arrange these 4 entities: $$4! = 24$$. Inside the block, A and B can switch seats in $$2! = 2$$ ways. So, total arrangements with A and B together: $$4! \times 2! = 24 \times 2 = 48$$. 4. **Calculate arrangements where A and B are NOT together:** Subtract the arrangements where A and B are together from the total arrangements. $$5! - (4! \times 2!) = 120 - 48 = 72$$. 5. **Answer:** There are $$72$$ seating arrangements where A and B are not sitting next to each other.