Question 1193717
**A. No Restriction**

* **Arrange the 14 people in a line:** 14! ways
* **Account for circular arrangements:** Divide by 14 to account for rotations (since any position can be considered the "start")
* **Number of ways:** 14! / 14 = 13! = 6,227,020,800 ways

**B. Girls and Boys Alternate**

* **Arrange the boys in a circle:** 6! ways (circular arrangement)
* **Place the girls in the 7 spaces between the boys:** 7! ways
* **Number of ways:** 6! * 7! = 50,400 * 5,040 = 254,016,000 ways

**C. 5 Particular Girls Must Sit Together**

* **Treat the 5 girls as a single unit:** Now we have 9 entities to arrange (6 boys + 4 units: 4 individual girls and 1 group of 5 girls)
* **Arrange the 9 entities in a circle:** 8! ways
* **Arrange the 5 girls within their group:** 5! ways
* **Number of ways:** 8! * 5! = 40,320 * 120 = 4,838,400 ways

**D. 5 Particular Girls Must Not Sit Together**

* **Find the total number of arrangements (from part A):** 13! ways
* **Find the number of arrangements where the 5 girls sit together (from part C):** 4,838,400 ways
* **Number of ways where 5 girls do not sit together:** 13! - 4,838,400 = 6,227,020,800 - 4,838,400 = 6,178,636,800 ways

**E. All the Girls Must Sit Together**

* **Treat the 7 girls as a single unit:** Now we have 8 entities to arrange (7 boys + 1 group of 7 girls)
* **Arrange the 8 entities in a circle:** 7! ways
* **Arrange the 7 girls within their group:** 7! ways
* **Number of ways:** 7! * 7! = 5,040 * 5,040 = 25,401,600 ways

I hope this helps! Let me know if you have any other questions.