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Form 5 Permutations and Combinations Practice

Form 5 · 60 practice questions with instant marking and worked solutions.

Permutations and Combinations is an important part of the Form 5 mathematics curriculum. This topic develops the multiplication principle, permutations, combinations and avoiding duplicates. Begin by identifying what every number, symbol, diagram and unit represents. Translate the information into a mathematical relationship before calculating. Each line of working should have a purpose. Finally use estimation, substitution, an inverse operation, or a second representation to check the result. This habit separates a misunderstanding from a copying error and makes correction more useful.

A reliable approach to Permutations and Combinations has four stages. First, mark the given information and the quantity required. Second, choose an appropriate representation, property or formula. Third, carry out the operations in a clear order while retaining signs and units. Fourth, return to the original context and decide whether the result is possible. For a graph or table, read the title, scale and labels before using a value. For algebra, inspect brackets, negative signs and the order of operations before simplifying.

The 60 original questions in this topic use a shared mathematical specification in Chinese and English. Difficulty rises from level 1 to level 5. Early questions establish the language and direct skills of Permutations and Combinations; middle questions add two-step reasoning, conversions and common variations; later questions connect ideas and require a deliberate check. In multiple-choice work, calculate before looking at the choices. In fill-in and short-answer work, retain decisive steps. For true-or-false statements, support the decision with a calculation or counterexample. 8 of them are DSE Paper 1 style structured questions answered in parts (a), (b), (c), where later parts build on earlier ones.

Effective revision is more than recording a score. After completing the Permutations and Combinations set, group errors by concept tags and decide whether each problem arose from interpretation, forming an expression, calculation, or presenting the answer. Retry the item on a later day without opening the solution, then test the same method with a different value. If the same tag appears repeatedly, return to the explanation and rebuild the concept. If signs or units are the issue, use a fixed pre-submission checklist.

Permutations and Combinations should also connect with the rest of mathematics. Number and algebra supply calculation tools; measurement and geometry give spatial meaning; data handling asks what a result means. A learner who can explain aloud why every step is valid has moved beyond imitation. Estimate a reasonable range, complete an exact solution, and compare the two. Agreement between context, estimate and exact answer is strong evidence that the solution is sound.

This batch adds 20 questions, bringing the topic to 60. The new questions cover controlled variations of different concepts, with paired bilingual conditions, hints, worked solutions and reproducible checks.

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  1. 1 Basic
    In Permutations and Combinations, calculate $3\times 6$. Choose the correct answer.
  2. 2 Basic
    In Permutations and Combinations, calculate $4\times 7$. Give the numerical value only.

  3. 3 Basic
    In Permutations and Combinations, calculate $5\times 3$. Show the essential working and give the numerical value.

  4. 4 Basic
    In Permutations and Combinations, the value of $6\times 4$ is 25. Is this statement true?
  5. 5 Basic
    From 5 different students, 3 are chosen to fill 3 different posts. Find the number of ways. Choose the correct answer.
  6. 6 Basic
    4 books are chosen from 10 different books. Find the number of selections. Give the answer only.

  7. 7 Basic
    4 people stand in a row and two particular people must stand together. Find the number of arrangements. Show the essential working and give the answer.

  8. 8 Basic
    A committee is formed by choosing 2 of 5 boys and 2 of 7 girls. The number of possible committees is 31. Is this true?
  9. 9 Basic
    From 8 different students, 4 are chosen to fill 4 different posts. Find the number of ways. Choose the correct answer.
  10. 10 Basic
    3 books are chosen from 7 different books. Find the number of selections. Give the answer only.

  11. 11 Basic
    7 people stand in a row and two particular people must stand together. Find the number of arrangements. Show the essential working and give the answer.

  12. 12 Basic
    A committee is formed by choosing 2 of 6 boys and 2 of 7 girls. The number of possible committees is 36. Is this true?
  13. 13 Standard
    In Permutations and Combinations, calculate $2\times 5$. Choose the correct answer.
  14. 14 Standard
    In Permutations and Combinations, calculate $3\times 6$. Give the numerical value only.

  15. 15 Standard
    In Permutations and Combinations, calculate $4\times 7$. Show the essential working and give the numerical value.

  16. 16 Standard
    In Permutations and Combinations, the value of $5\times 3$ is 15. Is this statement true?
  17. 17 Standard
    A class has 9 students, 4 of whom are monitor candidates. 3 students are chosen. Find the number of selections including at least one candidate. Choose the correct answer.
  18. 18 Standard
    From 10 different students, 3 are chosen to fill 3 different posts. Find the number of ways. Give the answer only.

  19. 19 Standard
    5 books are chosen from 11 different books. Find the number of selections. Show the essential working and give the answer.

  20. 20 Standard DSE long question
    A group of 4 is chosen from 7 boys and 3 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 1 girl.
    3. (c) Find the number of ways with at least one girl.
  21. 21 Standard
    A class has 9 students, 3 of whom are monitor candidates. 4 students are chosen. Find the number of selections including at least one candidate. Choose the correct answer.
  22. 22 Standard
    From 5 different students, 4 are chosen to fill 4 different posts. Find the number of ways. Give the answer only.

  23. 23 Standard
    2 books are chosen from 12 different books. Find the number of selections. Show the essential working and give the answer.

  24. 24 Standard DSE long question
    A group of 3 is chosen from 4 boys and 6 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 2 girls.
    3. (c) Find the number of ways with at least one girl.
  25. 25 Advanced
    In Permutations and Combinations, calculate $6\times 4$. Choose the correct answer.
  26. 26 Advanced
    In Permutations and Combinations, calculate $2\times 5$. Give the numerical value only.

  27. 27 Advanced
    In Permutations and Combinations, calculate $3\times 6$. Show the essential working and give the numerical value.

  28. 28 Advanced
    In Permutations and Combinations, the value of $4\times 7$ is 29. Is this statement true?
  29. 29 Advanced
    5 people stand in a row and two particular people must stand together. Find the number of arrangements. Choose the correct answer.
  30. 30 Advanced
    A committee is formed by choosing 3 of 5 boys and 1 of 6 girls. Find the number of possible committees. Give the answer only.

  31. 31 Advanced
    A class has 8 students, 5 of whom are monitor candidates. 4 students are chosen. The number of selections including at least one candidate is 70. Is this true?
  32. 32 Advanced DSE long question
    A bag has 5 red and 4 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  33. 33 Advanced
    6 people stand in a row and two particular people must stand together. Find the number of arrangements. Choose the correct answer.
  34. 34 Advanced
    A committee is formed by choosing 2 of 5 boys and 2 of 4 girls. Find the number of possible committees. Give the answer only.

  35. 35 Advanced
    A class has 8 students, 3 of whom are monitor candidates. 3 students are chosen. The number of selections including at least one candidate is 46. Is this true?
  36. 36 Advanced DSE long question
    A bag has 6 red and 4 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  37. 37 Challenging
    In Permutations and Combinations, calculate $5\times 3$. Choose the correct answer.
  38. 38 Challenging
    In Permutations and Combinations, calculate $6\times 4$. Give the numerical value only.

  39. 39 Challenging
    In Permutations and Combinations, calculate $2\times 5$. Show the essential working and give the numerical value.

  40. 40 Challenging
    In Permutations and Combinations, the value of $3\times 6$ is 18. Is this statement true?
  41. 41 Challenging
    From 6 different students, 3 are chosen to fill 3 different posts. Find the number of ways. Choose the correct answer.
  42. 42 Challenging
    3 books are chosen from 12 different books. Find the number of selections. Show the essential working and give the answer.

  43. 43 Challenging
    7 people stand in a row and two particular people must stand together. The number of arrangements is 720. Is this true?
  44. 44 Challenging DSE long question
    A group of 3 is chosen from 4 boys and 4 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 2 girls.
    3. (c) Find the number of ways with at least one girl.
  45. 45 Challenging
    From 7 different students, 4 are chosen to fill 4 different posts. Find the number of ways. Choose the correct answer.
  46. 46 Challenging
    5 books are chosen from 9 different books. Find the number of selections. Show the essential working and give the answer.

  47. 47 Challenging
    5 people stand in a row and two particular people must stand together. The number of arrangements is 24. Is this true?
  48. 48 Challenging DSE long question
    A group of 4 is chosen from 4 boys and 4 girls. Answer each part; give the answer only.
    1. (a) Find the total number of ways.
    2. (b) Find the number of ways with exactly 1 girl.
    3. (c) Find the number of ways with at least one girl.
  49. 49 Elite
    In Permutations and Combinations, calculate $4\times 7$. Choose the correct answer.
  50. 50 Elite
    In Permutations and Combinations, calculate $5\times 3$. Give the numerical value only.

  51. 51 Elite
    In Permutations and Combinations, calculate $6\times 4$. Show the essential working and give the numerical value.

  52. 52 Elite
    In Permutations and Combinations, the value of $2\times 5$ is 11. Is this statement true?
  53. 53 Elite
    A committee is formed by choosing 2 of 6 boys and 2 of 3 girls. Find the number of possible committees. Give the answer only.

  54. 54 Elite
    A class has 12 students, 3 of whom are monitor candidates. 3 students are chosen. Find the number of selections including at least one candidate. Show the essential working and give the answer.

  55. 55 Elite
    From 9 different students, 3 are chosen to fill 3 different posts. The number of ways is 504. Is this true?
  56. 56 Elite DSE long question
    A bag has 5 red and 2 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).
  57. 57 Elite
    A committee is formed by choosing 3 of 4 boys and 1 of 7 girls. Find the number of possible committees. Give the answer only.

  58. 58 Elite
    A class has 11 students, 5 of whom are monitor candidates. 3 students are chosen. Find the number of selections including at least one candidate. Show the essential working and give the answer.

  59. 59 Elite
    From 5 different students, 3 are chosen to fill 3 different posts. The number of ways is 60. Is this true?
  60. 60 Elite DSE long question
    A bag has 7 red and 2 blue balls. Two are drawn without replacement. Answer each part; give the answer only.
    1. (a) Find the probability that the first is red (as a fraction in lowest terms).
    2. (b) Find the probability that both are red (as a fraction in lowest terms).
    3. (c) Find the probability of at least one blue ball (as a fraction in lowest terms).

The report lists what you got wrong and why. Only you can see it.

Common mistakes

Common mistakes in Permutations and Combinations

This topic needs particular care with the multiplication principle, permutations, combinations and avoiding duplicates. The first mistake is applying a remembered rule before reading all conditions. The second is compressing working so much that a sign, bracket, decimal point or unit disappears. The third is accepting an impossible answer without checking: for example, a probability above one, a negative physical length, an angle outside the geometric constraints, or a result far from the estimate. Separate the conditions, model, calculation and check, and finish by stating what the answer represents.

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