1. **Problem statement:** Find the number of ways to seat five couples in a row under different conditions. 2. **General formula:** The number of ways to arrange $n$ distinct people in a row is $n!$. --- ### a. Without restrictions 3. There are 10 people total (5 couples), so the number of ways is: $$10! = 3628800$$ --- ### b. If each couple is seated together 4. Treat each couple as a single unit. There are 5 units to arrange: $$5! = 120$$ 5. Each couple can be arranged internally in $2!$ ways (man-woman or woman-man): $$2^5 = 32$$ 6. Total arrangements: $$5! \times 2^5 = 120 \times 32 = 3840$$ --- ### c. If all females are seated together 7. Treat all 5 females as one block. Along with 5 males, total units to arrange: $$6! = 720$$ 8. Females inside the block can be arranged in: $$5! = 120$$ 9. Total arrangements: $$6! \times 5! = 720 \times 120 = 86400$$ --- ### d. If all males are seated together as well as females are seated together 10. Treat males as one block and females as another block. These 2 blocks can be arranged in: $$2! = 2$$ 11. Inside each block, arrange 5 people: $$5! = 120$$ 12. Total arrangements: $$2! \times 5! \times 5! = 2 \times 120 \times 120 = 28800$$ --- ### e. If all males are seated to the right of all females 13. Females occupy the first 5 seats, males the last 5 seats. 14. Arrange females among themselves: $$5! = 120$$ 15. Arrange males among themselves: $$5! = 120$$ 16. Total arrangements: $$5! \times 5! = 120 \times 120 = 14400$$