1. **Problem:** Given $n=6$ and $r=3$, determine which expressions about permutations $P(6,3)$ and combinations $C(6,3)$ are true. 2. **Formulas:** - Permutation: $$P(n,r) = \frac{n!}{(n-r)!}$$ - Combination: $$C(n,r) = \frac{n!}{r!(n-r)!}$$ 3. **Calculate $P(6,3)$:** $$P(6,3) = \frac{6!}{(6-3)!} = \frac{6!}{3!} = \frac{720}{6} = 120$$ 4. **Calculate $C(6,3)$:** $$C(6,3) = \frac{6!}{3!3!} = \frac{720}{6 \times 6} = \frac{720}{36} = 20$$ 5. **Check expressions:** - I. $P(6,3) = 120$ is **true**. - II. $C(6,3) = 18$ is **false** (correct value is 20). - III. $P(6,3) = 20$ is **false**. 6. **Answer for question 18:** Only statement I is true, so none of the options exactly match. Since options are combinations of statements, the closest correct is option B (I and III) but III is false, so none are fully correct. The problem likely expects option A (I and II) but II is false. So the correct conclusion is only I is true. --- 7. **Problem 19:** How many different groups of 13 people can be formed from 15? 8. **Formula:** Combination $$C(15,13) = \frac{15!}{13!2!} = \frac{15 \times 14}{2} = 105$$ 9. **Answer:** 105 (option B) --- 10. **Problem 20:** Which situation uses permutation notation $_nP_r$? 11. **Explanation:** Permutations count ordered arrangements. - A: Drawing numbers (order matters) → permutation - B: Choosing players (order does not matter) → combination - C: Forming committee (order does not matter) → combination - D: Four-digit lock (order matters) → permutation 12. **Answer:** A and D use permutation notation. --- 13. **Problem 21:** From 10 girls and 8 boys, how many groups of 3 girls and 2 boys? 14. **Calculation:** $$C(10,3) = \frac{10!}{3!7!} = 120$$ $$C(8,2) = \frac{8!}{2!6!} = 28$$ Total groups = $120 \times 28 = 3360$ 15. **Answer:** 3360 (option A) --- 16. **Problem 22:** Number of ways to arrange 7 potted plants in a row? 17. **Calculation:** $$7! = 5040$$ 18. **Answer:** 5040 (option A) --- 19. **Problem 23:** Number of plate numbers from symbols GUD 272? 20. **Symbols:** 6 characters with '2' repeated twice. 21. **Formula:** $$\frac{6!}{2!} = \frac{720}{2} = 360$$ 22. **Answer:** 360 (option B) --- 23. **Problem 24:** Ways to seat 7 people around a round table so two specific people do NOT sit together? 24. **Total arrangements:** $$(7-1)! = 6! = 720$$ 25. **Arrangements with two specific people together:** Treat them as one unit: $$(6-1)! \times 2! = 5! \times 2 = 120 \times 2 = 240$$ 26. **Arrangements where they do NOT sit together:** $$720 - 240 = 480$$ 27. **Answer:** Not listed exactly; closest is option D (600) but correct is 480. --- 28. **Problem 25:** Number of distinct circular permutations for $n$ distinct objects? 29. **Formula:** $$(n-1)!$$ 30. **Answer:** (n-1)! (option B) --- 31. **Problem 26:** 5 people around table, two insist sitting together? 32. **Calculation:** Treat two as one unit: $$(5-1)! \times 2! = 4! \times 2 = 24 \times 2 = 48$$ 33. **Answer:** 48 (option A) --- 34. **Problem 27:** Circular permutation of 9 people, two seated together? 35. **Calculation:** $$(9-1)! = 8! = 40320$$ $$\text{With two together} = 2! \times (8-1)! = 2 \times 7! = 2 \times 5040 = 10080$$ 36. **Answer:** 2 x 7! (option A) --- 37. **Problem 28:** Number of ways to arrange 6 people around table so two never sit together? 38. **Total:** $$5! = 120$$ 39. **Together:** $$2 \times 4! = 2 \times 24 = 48$$ 40. **Never together:** $$120 - 48 = 72$$ 41. **Answer:** 5! - 2 x 4! (option B) --- 42. **Problem 29:** Circular permutations of 8 beads on necklace where rotations count same? 43. **Formula:** $$\frac{(n-1)!}{2} = \frac{7!}{2}$$ 44. **Answer:** (n-1)!/2 (option C) --- 45. **Problem 30:** Permutations of 6 people dining around circular table? 46. **Formula:** $$(6-1)! = 5! = 120$$ 47. **Answer:** 120 (option B)