Class 9 Maths Combinatorics Notes

Class 9 Maths Combinatorics Notes

Chapter 5 — Combinatorics

1. Fundamental Principle of Counting

Many difficult counting problems become simple when broken into stages.

Multiplication / AND rule

If one step has m choices and another independent step has n choices:

Total = m × n

Use this when both stages happen together.

Addition / OR rule

If choices come from mutually exclusive alternatives:

Total = m + n

Use this when you choose one alternative OR another.

2. Tree Diagrams

A tree diagram shows choices as branches.

It is especially useful for small problems because every endpoint represents one possible outcome.

For larger problems, multiplication gives the same count more efficiently.

3. Factorial

For a positive integer n:

n! = n(n−1)(n−2)...3×2×1

Examples:

3! = 6

5! = 120

Also:

0! = 1

Factorials are fundamental to arrangements.

4. Permutations

A permutation is an arrangement, so order matters.

The number of arrangements of n distinct objects taken r at a time is:

ⁿPᵣ = n!/(n−r)!

Use permutations when changing the order produces a different outcome.

5. Combinations

A combination is a selection, so order does not matter.

The number of ways of selecting r objects from n is:

ⁿCᵣ = n!/[r!(n−r)!]

A useful relationship is:

ⁿPᵣ = ⁿCᵣ × r!

6. Choosing the Correct Method

Ask:

Does order matter?

  • Yes → permutation
  • No → combination

Then ask:

Are there separate alternatives?

  • mutually exclusive choices → addition
  • successive choices → multiplication

This decision process is often more important than memorising formulas.

7. Applications

The chapter applies counting ideas to:

  • passwords
  • arranging people or objects
  • forming teams
  • selecting examination questions
  • scheduling tasks

Exam Focus

Before writing a formula, identify the nature of the problem:
AND → multiply, OR → add, arrange → permutation, select → combination.

Chapter 5 — Combinatorics: Complete Question Bank

A. Multiple Choice Questions (MCQs)

1. Combinatorics is mainly concerned with:

A. Measuring angles
B. Counting and arranging possibilities
C. Solving quadratic equations
D. Measuring areas

2. If one task can be completed in 4 ways and a second independent task in 5 ways, the number of ways to complete both tasks is:

A. 9
B. 20
C. 1
D. 25

3. The multiplication rule is commonly called the:

A. OR rule
B. AND rule
C. Selection rule
D. Factorial rule

4. The addition rule is used when choices are:

A. Successive
B. Independent and simultaneous
C. Mutually exclusive
D. Repeated

5. A tree diagram is particularly useful for:

A. Showing possible outcomes
B. Measuring distances
C. Finding slopes
D. Drawing circles

6. The value of 5! is:

A. 25
B. 60
C. 100
D. 120

7. Which of the following is equal to 0!?

A. 0
B. 1
C. Undefined
D. −1

8. In a permutation:

A. Order does not matter
B. Order matters
C. Only repetition matters
D. Selection is impossible

9. The number of permutations of n distinct objects taken r at a time is:

A. n!/[r!(n−r)!]
B. n!/(n−r)!
C. n!/r!
D. (n−r)!/n!

10. In a combination:

A. Order matters
B. Order does not matter
C. Objects must be arranged
D. Objects cannot be selected

11. The formula for selecting r objects from n objects is:

A. ⁿPᵣ
B. ⁿCᵣ
C. n+r
D. n−r

12. Which situation is best represented by a combination?

A. Arranging students in seats
B. Assigning first, second and third positions
C. Selecting members for a committee
D. Creating a sequence of letters

13. ⁶P₂ equals:

A. 12
B. 15
C. 30
D. 36

14. ⁶C₂ equals:

A. 12
B. 15
C. 20
D. 30

15. The relationship between permutations and combinations is:

A. ⁿPᵣ = ⁿCᵣ/r!
B. ⁿPᵣ = ⁿCᵣ × r!
C. ⁿPᵣ = ⁿCᵣ + r!
D. ⁿPᵣ = ⁿCᵣ − r!

B. Fill in the Blanks

16. Combinatorics is the study of counting, arrangements and __________.

17. The Fundamental Principle of Counting has two basic rules: multiplication and __________.

18. The multiplication rule is used when multiple steps must occur __________.

19. The addition rule is used for mutually __________ choices.

20. A diagram that branches at each decision point is called a __________ diagram.

21. The symbol used for factorial is __________.

22. n! = n × (n−1) × (n−2) × ... × __________.

23. 0! = __________.

24. A permutation is an __________ of objects in which order matters.

25. A combination is a __________ in which order does not matter.

26. The number of permutations of n objects taken r at a time is denoted by __________.

27. The number of combinations of n objects taken r at a time is denoted by __________.

28. ⁿC₀ = __________.

29. ⁿCₙ = __________.

30. ⁿCᵣ = ⁿC__________.

C. True or False

31. The multiplication rule is used for successive choices.

32. The addition rule can be used for overlapping choices without adjustment.

33. A tree diagram can be used to verify a counting result.

34. 4! = 24.

35. 1! = 0.

36. In permutations, changing the order generally produces a different arrangement.

37. In combinations, changing the order creates a new selection.

38. A committee-selection problem is generally a combination problem.

39. ⁵P₂ = 10.

40. ⁵C₂ = 10.

41. ⁿPᵣ = ⁿCᵣ × r!.

42. ⁿCᵣ and ⁿCₙ₋ᵣ have the same value.

D. Match the Following

Column AColumn B
43. AND rulea. Order does not matter
44. OR ruleb. n!/(n−r)!
45. Permutationc. Multiply
46. Combinationd. Add
47. Factoriale. n(n−1)...1

E. Fundamental Principle of Counting

48. A student has 3 shirts and 4 pairs of trousers. How many different outfits can be formed?

49. A café offers 5 sandwiches and 3 drinks. If one sandwich and one drink are selected, how many choices are possible?

50. A test contains 4 questions, each having 3 possible answers. How many different answer patterns are possible?

51. A code consists of one letter followed by one digit. If 5 letters and 6 digits are available, how many codes can be formed?

52. A student can travel to school by 4 bus routes or 2 train routes. If exactly one route is chosen, how many choices are available?

53. A shop offers 4 types of notebooks, 3 types of pens and 2 types of folders. If one of each is purchased, find the number of possible selections.

54. Explain when you would use addition rather than multiplication in a counting problem.

F. Tree Diagram Questions

55. A student has 2 choices of breakfast and 3 choices of juice. Describe how a tree diagram can represent all possible combinations.

56. A coin is tossed twice. How many possible outcomes are there? Represent the logic using a tree structure.

57. A code uses either one of 3 letters followed by one of 2 digits. How many codes are possible?

58. Explain why the number of endpoints in a complete tree diagram represents the number of possible outcomes.

G. Factorials

59. Evaluate 4!.

60. Evaluate 6!.

61. Evaluate 7!.

62. Simplify:

8!/6!

63. Simplify:

9!/7!

64. Express 6! as a product of consecutive positive integers.

65. If n! = 120, find n.

66. Explain why 0! is defined as 1 in combinatorial calculations.

H. Permutations

67. Find ⁵P₂.

68. Find ⁶P₃.

69. Find ⁷P₂.

70. In how many ways can 4 different books be arranged on a shelf?

71. In how many ways can 3 students be selected and assigned the positions of captain, vice-captain and secretary from 8 students?

72. How many 3-letter arrangements can be made from 5 distinct letters if repetition is not allowed?

73. In how many ways can 5 different people stand in a line?

74. How many 4-digit numbers can be formed using four distinct selected digits when their order can be changed freely?

75. Explain why arranging 3 students in a row is a permutation rather than a combination.

I. Combinations

76. Find ⁵C₂.

77. Find ⁷C₃.

78. Find ⁸C₂.

79. In how many ways can 3 students be selected from a class of 10?

80. In how many ways can a committee of 4 people be selected from 9 people?

81. From 8 players, how many different groups of 2 players can be selected?

82. A teacher asks students to choose 5 questions from a set of 8. In how many ways can the questions be selected?

83. Explain why selecting a team of students is usually a combination problem.

J. Permutation or Combination?

Identify the appropriate method.

84. Arranging 6 books on a shelf.

85. Selecting 3 students for a committee.

86. Choosing 2 toppings from 7 available toppings.

87. Assigning gold, silver and bronze medals.

88. Selecting 5 questions from 10 questions.

89. Arranging 4 different photographs in a row.

90. Choosing 3 players for a team.

K. Application-Based Questions

91. A password contains 4 positions, and each position can contain one of 6 symbols. Repetition is allowed. How many passwords are possible?

92. A school has 6 clubs, 4 sports activities and 3 music activities. A student must choose exactly one activity. How many choices are possible?

93. A restaurant offers 5 starters, 6 main dishes and 4 desserts. A customer selects one from each category. Find the number of possible meals.

94. From 7 students, a captain and vice-captain are to be chosen. How many possible assignments are there?

95. From 7 students, 2 representatives are to be chosen without assigning different positions. How many possible pairs are there?

96. A set of 10 points is given in a plane, with no three points on the same straight line. How many different line segments can be determined by choosing pairs of points?

97. A polygon has 8 vertices. How many pairs of vertices can be selected?

98. A team must contain 3 boys chosen from 6 boys and 2 girls chosen from 5 girls. How many different teams can be formed?

L. Assertion–Reason Questions

Choose:

A. Both Assertion and Reason are true, and Reason correctly explains Assertion.
B. Both are true, but Reason does not correctly explain Assertion.
C. Assertion is true, Reason is false.
D. Assertion is false, Reason is true.

99. Assertion: If two independent stages have 4 and 5 choices respectively, there are 20 ways to complete both stages.
Reason: The multiplication rule is used when both stages are required.

100. Assertion: Selecting a committee is generally a permutation problem.
Reason: The order in which committee members are selected does not change the committee.

101. Assertion: 5! = 120.
Reason: 5! = 5×4×3×2×1.

102. Assertion: ⁷C₂ = ⁷C₅.
Reason: Selecting 2 objects is complementary to leaving 5 objects unselected.

103. Assertion: A tree diagram can help check the number of outcomes in a small counting problem.
Reason: Each complete branch represents one possible outcome.

M. Higher-Order Thinking Questions

104. A student uses the addition rule to solve a problem involving choosing one shirt AND one pair of trousers. Explain the mistake.

105. A class has 8 boys and 6 girls. In how many ways can one boy AND one girl be selected?

106. From 8 students, a captain and vice-captain are chosen. Another group of 2 students is simply selected as representatives. Explain why the two questions require different methods.

107. Show using formulas that:

⁷P₂ = ⁷C₂ × 2!

108. A student calculates ⁸C₃ and ⁸C₅ separately. Without calculating both completely, explain why the answers must be equal.

109. There are 6 different books. In how many ways can 3 of them be selected and then arranged?

110. A school wants to select 3 students from 10 and then appoint one of those three as leader. Describe a suitable counting method.

111. A 4-digit code uses digits 0–9, with repetition allowed. How does the answer change if repetition is prohibited?

112. Explain why “AND” and “OR” are useful keywords when deciding which counting rule to apply.

N. Mixed Challenge Questions

113. How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 without repetition?

114. How many 3-member teams can be selected from 9 students?

115. In how many ways can 4 different books be arranged if a particular book must always occupy the first position?

116. From 5 boys and 4 girls, how many teams containing exactly 2 boys and 2 girls can be formed?

117. How many different arrangements of 4 distinct objects can be made by taking 2 at a time?

118. A student must select either 2 science projects from 5 options or 1 mathematics project from 4 options. How many possible selections are there?

Answer Key

MCQs

  1. B
  2. B
  3. B
  4. C
  5. A
  6. D
  7. B
  8. B
  9. B
  10. B
  11. B
  12. C
  13. C
  14. B
  15. B

Fill in the Blanks

  1. selections
  2. addition
  3. together
  4. exclusive
  5. tree
  6. !
  7. 1
  8. 1
  9. arrangement
  10. selection
  11. ⁿPᵣ
  12. ⁿCᵣ
  13. 0
  14. 1
  15. n−r

True/False

  1. True
  2. False
  3. True
  4. True
  5. False
  6. True
  7. False
  8. True
  9. False
  10. True
  11. True
  12. True

Match

43–c
44–d
45–b
46–a
47–e

Numerical Answers

  1. 12
  2. 15
  3. 81
  4. 30
  5. 6
  6. 24
  7. 24
  8. 720
  9. 5040
  10. 56
  11. 72
  12. 5
  13. 20
  14. 120
  15. 42
  16. 24
  17. 336
  18. 60
  19. 120
  20. 10
  21. 35
  22. 28
  23. 120
  24. 126
  25. 28
  26. 56
  27. 1296
  28. 13
  29. 120
  30. 42
  31. 21
  32. 45
  33. 28
  34. 150

Permutation or Combination?

  1. Permutation
  2. Combination
  3. Combination
  4. Permutation
  5. Combination
  6. Permutation
  7. Combination

Assertion–Reason

  1. A
  2. D
  3. A
  4. A
  5. A

Challenge Answers

  1. 48
  2. ⁷P₂ = 7×6 = 42; ⁷C₂×2! = 21×2 = 42
  3. 120
  4. 60
  5. 84
  6. 6
  7. 60
  8. 12
  9. 14

Quick Revision Formula Sheet

Fundamental Principle of Counting

AND → Multiply

If there are m ways for one stage and n ways for another:

Total = m × n

OR → Add

For mutually exclusive alternatives:

Total = m + n

Factorial

n! = n(n−1)(n−2)...2×1

0! = 1

Permutation

ⁿPᵣ = n!/(n−r)!

Combination

ⁿCᵣ = n!/[r!(n−r)!]

Important Relationship

ⁿPᵣ = ⁿCᵣ × r!

Symmetry of Combinations

ⁿCᵣ = ⁿCₙ₋ᵣ