Question: Find the number of different permutations:
( 1 ) 6 men and 2 women stand in a line, 2 women next to each other.
( 2 ) 6 men and 2 women stand in a line, 2 women cannot be next to each other.
( 3 ) 4 men and 4 women stand in a line, people of the same sex next to each other.
( 4 ) 4 men and 4 women stand in a line. People of the same sex cannot be next to each other.
1. **Problem 1:** Find the number of permutations where 6 men and 2 women stand in a line with the 2 women next to each other.
2. **Formula and explanation:** Treat the 2 women as a single block since they must be together.
- Number of entities to arrange: 6 men + 1 block of 2 women = 7 entities.
- Number of ways to arrange these 7 entities: $$7!$$
- Number of ways to arrange the 2 women inside their block: $$2!$$
3. **Calculation:**
$$\text{Total permutations} = 7! \times 2! = 5040 \times 2 = 10080$$
4. **Problem 2:** Find the number of permutations where 6 men and 2 women stand in a line with the 2 women NOT next to each other.
5. **Formula and explanation:** Total permutations minus permutations where women are together.
- Total permutations without restriction: $$8!$$ (6 men + 2 women)
- Permutations with women together (from Problem 1): $$10080$$
6. **Calculation:**
$$\text{Total permutations} = 8! - 10080 = 40320 - 10080 = 30240$$
7. **Problem 3:** Find the number of permutations where 4 men and 4 women stand in a line, and people of the same sex are next to each other.
8. **Formula and explanation:** This means all men are together and all women are together.
- Treat men as one block and women as another block.
- Number of ways to arrange the two blocks: $$2!$$
- Number of ways to arrange men inside their block: $$4!$$
- Number of ways to arrange women inside their block: $$4!$$
9. **Calculation:**
$$\text{Total permutations} = 2! \times 4! \times 4! = 2 \times 24 \times 24 = 1152$$
10. **Problem 4:** Find the number of permutations where 4 men and 4 women stand in a line, and people of the same sex cannot be next to each other.
11. **Formula and explanation:** Men and women must alternate.
- Since numbers are equal, arrangements must be alternating: M W M W M W M W or W M W M W M W M.
- Number of ways to arrange men in their positions: $$4!$$
- Number of ways to arrange women in their positions: $$4!$$
- Number of ways to choose the pattern (start with man or woman): $$2!$$
12. **Calculation:**
$$\text{Total permutations} = 2! \times 4! \times 4! = 2 \times 24 \times 24 = 1152$$
**Final answers:**
1) $$10080$$
2) $$30240$$
3) $$1152$$
4) $$1152$$