1. **Problem:** There are 5 towns and we want to arrange their pictures in a row. How many ways can they be arranged? 2. **Formula:** The number of ways to arrange $n$ distinct objects in a row is given by the permutation formula: $$P_n = n!$$ where $n! = n \times (n-1) \times (n-2) \times \cdots \times 1$. 3. **Solution for 5 towns:** $$5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$$ So, there are 120 ways to arrange 5 towns in a row. --- 4. **Problem:** 4 twins want to be together. 5. **Part A:** 4 twins must be together. 6. **Explanation:** Treat the 4 twins as a single unit. Then arrange this unit along with any other individuals. 7. **Number of ways to arrange:** - Treat 4 twins as 1 unit + other individuals (if any). - Number of ways to arrange the unit and others = factorial of total units. - Number of ways to arrange the 4 twins inside the unit = $4!$. 8. **Part B:** 4 twins tossed together (assuming no restriction). 9. **Number of ways:** $4! = 24$ ways. --- 10. **Problem:** 10 persons take dinner at a round table. How many ways can they be seated? 11. **Formula for circular permutation:** $$P_{circular} = (n-1)!$$ because rotations are considered the same arrangement. 12. **Solution:** $$9! = 362880$$ ways. --- 13. **Problem:** Find permutations of the word MISSISSIPPI. 14. **Explanation:** The word has 11 letters with repetitions: - M:1 - I:4 - S:4 - P:2 15. **Formula for permutations with repeated letters:** $$\frac{n!}{n_1! \times n_2! \times \cdots}$$ 16. **Calculation:** $$\frac{11!}{1! \times 4! \times 4! \times 2!} = \frac{39916800}{1 \times 24 \times 24 \times 2} = 34650$$ 17. **Five possible permutations:** MISSISSIPPI, MISPISSIPI, ISSMIPISPI, PIMISSISPI, SIPMISSIPI (examples). --- 18. **Problem:** Possible permutations of tossing a coin. 19. **Explanation:** Each toss has 2 outcomes: Head (H) or Tail (T). 20. **For 1 toss:** 2 permutations: H, T. 21. **Diagram:** - Toss 1: H or T --- 22. **Problem:** Letters of the word COPYING arranged so that the letters of COPYING must be together. 23. **Explanation:** Treat the word COPYING as a single unit. 24. **Number of letters:** 7 letters, so only 1 unit. 25. **Number of ways:** Since all letters are together, only 1 way to arrange the unit. --- 26. **Problem:** Mathematics club with 9 boys and 7 girls. Form a team of 5 members with at least 2 girls. 27. **Formula:** Use combinations: $$C(n, r) = \frac{n!}{r!(n-r)!}$$ 28. **Calculate:** Sum of teams with 2, 3, 4, or 5 girls. - For 2 girls: $C(7,2) \times C(9,3)$ - For 3 girls: $C(7,3) \times C(9,2)$ - For 4 girls: $C(7,4) \times C(9,1)$ - For 5 girls: $C(7,5) \times C(9,0)$ 29. **Total ways:** $$C(7,2)C(9,3) + C(7,3)C(9,2) + C(7,4)C(9,1) + C(7,5)C(9,0)$$ --- 30. **Problem:** Permutation of 4 colors (yellow, blue, red, pink), 3 shoes (nike, adidas, converse), and 2 socks (small, large). 31. **Number of ways:** $$4! \times 3! \times 2! = 24 \times 6 \times 2 = 288$$ --- 32. **Problem:** 4 coins tossed together, write possible permutations. 33. **Number of outcomes:** $$2^4 = 16$$ Examples: HHHH, HHHT, HHTH, HTHH, THHH, etc. --- 34. **Problem:** Four questions answered True or False, possible permutations? 35. **Number of outcomes:** $$2^4 = 16$$ --- 36. **Problem:** Coin tossed 4 times, permutations with at most 1 tail. 37. **Calculate:** - 0 tails: 1 way (HHHH) - 1 tail: $C(4,1) = 4$ ways 38. **Total:** $1 + 4 = 5$ ways. --- 39. **Problem:** Canteen choices: 2 rice, 3 viand, 2 drinks. 40. **Permutation of fried rice, fried chicken, and drink:** $$1 \times 1 \times 2 = 2$$ 41. **Permutation of fried rice, fried chicken, tinolang manok, sinugbang bangus, and drinks:** $$1 \times 3 \times 2 = 6$$ --- 42. **Problem:** Different ways to arrange letters of "yellow nike" with small size sock. 43. **Explanation:** Treat as permutation of letters plus sock. 44. **Number of letters:** 10 letters (yellownike), all distinct. 45. **Number of ways:** $$10! \times 1 = 3628800$$ --- 46. **Problem:** Coin tossed 5 times, number of possible outcomes with at most 2 tails. 47. **Calculate:** - 0 tails: $C(5,0) = 1$ - 1 tail: $C(5,1) = 5$ - 2 tails: $C(5,2) = 10$ 48. **Total:** $1 + 5 + 10 = 16$ ways. --- 49. **Problem:** Rolling a die and tossing a coin, possible permutations with even number and tail. 50. **Even numbers on die:** 2, 4, 6 51. **Coin outcome:** Tail 52. **Number of permutations:** $$3 \times 1 = 3$$ Possible outcomes: (2, T), (4, T), (6, T) --- **Summary:** - Permutations of 5 towns in a row: 120 - 4 twins together: treat as unit, multiply internal arrangements - Circular permutation of 10 persons: 9! - Permutations of MISSISSIPPI: 34650 - Coin toss permutations: $2^n$ - Teams with at least 2 girls: sum of combinations - Colors, shoes, socks permutations: product of factorials - Coin toss with at most k tails: sum of combinations - Rolling die and coin with conditions: multiply possibilities