1. **Problem statement:** We have 5 couples (10 people) to be seated in a row. We want to find the number of ways to arrange them under different conditions. 2. **General formula for permutations:** The number of ways to arrange $n$ distinct people in a row is $n!$. --- **a. Without restrictions:** - Total people: 10 - Number of ways: $$10! = 3628800$$ --- **b. If each couple is seated together:** - Treat each couple as a single unit: 5 units - Number of ways to arrange these 5 units: $$5!$$ - Each couple can be arranged internally in $$2!$$ ways - Total ways: $$5! \times (2!)^5 = 120 \times 32 = 3840$$ --- **c. If all females are seated together:** - Treat all 5 females as one block - Remaining 5 males are individual - Number of units to arrange: 6 (1 female block + 5 males) - Ways to arrange these units: $$6!$$ - Ways to arrange females inside the block: $$5!$$ - Total ways: $$6! \times 5! = 720 \times 120 = 86400$$ --- **d. If all males are seated together as well as females are seated together:** - Treat females as one block and males as another block - Number of units to arrange: 2 (female block and male block) - Ways to arrange these 2 blocks: $$2!$$ - Ways to arrange females inside their block: $$5!$$ - Ways to arrange males inside their block: $$5!$$ - Total ways: $$2! \times 5! \times 5! = 2 \times 120 \times 120 = 28800$$ --- **e. If all males are seated to the right of all females:** - Females occupy the first 5 seats, males the last 5 seats - Ways to arrange females among themselves: $$5!$$ - Ways to arrange males among themselves: $$5!$$ - Total ways: $$5! \times 5! = 120 \times 120 = 14400$$