Question 1193717: How many ways can 7 boys and 7 girls be seated at a round table if:
A. No restriction is imposed?
B. The girls and the boys are to occupy alternate seats?
C. 5 particular girls must sit together
D. 5 particular girls must not sit together?
E. All the girls must sit together?
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **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.
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