Permutation, Combination & Probability: Comprehensive Notes and 100 Practice Questions with Answers

 

Permutation, Combination & Probability: Comprehensive Notes and 100 Practice Questions with Answers

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1. Counting Principles

A. Fundamental Principle of Counting

• If an event can occur in $m$ ways and another in $n$ ways, both can occur in $m\times n$ ways.

• Example: If you have 3 shirts and 2 pants, you can dress in $3\times 2=6$ ways.

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2. Permutations

A. Definition

• Permutation is an arrangement of objects where the order matters146.

B. Formulae

• Without Repetition:
  $n{P}_r=\frac{n!}{(n-r)!}$

• With Repetition:
  $n^r$
  (Each of the $r$ positions can be filled in $n$ ways.)

C. Examples

• Arranging 3 books on a shelf: $3!=6$ ways.

• Arranging 2 out of 5 people in a line: $5P_2=5\times 4=20$ ways16.

• 5-digit number with no repeated digits: $10P_5=30,240$ ways2.

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3. Combinations

A. Definition

• Combination is a selection of objects where the order does NOT matter146.

B. Formulae

• Without Repetition:
  $n{C}_r=\frac{n!}{r!(n-r)!}$

• With Repetition:
  $\frac{(n+r-1)!}{r!(n-1)!}$

C. Examples

• Selecting 2 out of 5 people: $5C_2=10$ ways16.

• Forming a committee of 2 men and 1 woman from 5 men and 4 women: $5C_2\times 4C_1=10\times 4=40$ ways6.

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4. Difference Between Permutation and Combination

Permutation (Order matters)	Combination (Order does not matter)
Arranging people, digits, letters	Selecting people, menu, teams
E.g., 1st, 2nd, 3rd place in a race	E.g., picking any 3 winners
$nPr$ formula	$nCr$ formula

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5. Probability Basics

A. Definition

• Probability of an event = $\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

B. Using Permutations and Combinations in Probability

• Sample space: Count all possible outcomes (often using permutations or combinations).

• Favorable outcomes: Count outcomes that meet the required condition237.

C. Example

• Probability of guessing a 5-digit number with no repeats:
  Total possible = $10P_5$, Favorable = 1
  Probability = $\frac{1}{10P_5}$2.

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6. Step-by-Step Example: Probability with Permutations/Combinations

Example: What is the probability of selecting 2 red balls from a bag of 5 red and 3 blue balls?

• Total ways to select 2 balls: $8C_2=28$

• Ways to select 2 red balls: $5C_2=10$

• Probability = $\frac{10}{28}=\frac{5}{14}$38.

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7. Key Tips

• Use permutations when order/arrangement matters; combinations when only selection matters146.

• For probability, always clearly define the sample space and favorable outcomes using the correct counting method2357.

• Factorials: $n!=n\times (n-1)\times \mathrm{.}\mathrm{.}\mathrm{.}\times 1$

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100 Practice Questions with Answers and Explanations

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Counting Principles, Arrangements, and Selections

1. How many ways can 4 books be arranged on a shelf?
   Ans: $4!=24$

2. How many ways can you arrange 3 letters from A, B, C, D?
   Ans: $4P_3=24$

3. How many ways can you select 2 students from 6?
   Ans: $6C_2=15$

4. How many ways to arrange 5 people in a row?
   Ans: $5!=120$

5. How many ways to select a president and a secretary from 10 members?
   Ans: $10P_2=90$

6. How many ways to select 3 balls from 7?
   Ans: $7C_3=35$

7. How many ways to arrange the letters of the word 'MATH'?
   Ans: $4!=24$

8. How many ways to select 2 red and 1 blue ball from 4 red and 3 blue balls?
   Ans: $4C_2\times 3C_1=6\times 3=18$

9. How many 3-digit numbers can be formed from 1, 2, 3, 4 without repetition?
   Ans: $4P_3=24$

10. How many ways to arrange 5 people around a round table?
    Ans: $(5-1)!=24$

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Permutations (Order Matters)

11. How many ways to arrange 3 out of 8 books?
    Ans: $8P_3=336$

12. How many 4-letter words with no repetition from ABCDE?
    Ans: $5P_4=120$

13. How many ways to seat 4 boys and 3 girls in a row so that boys and girls alternate?
    Ans: Boys: $4!$, Girls: $3!$, Arrangements: 2 (BGBGBGB or GBGBGBG).
    $2\times 4!\times 3!=288$

14. How many ways to arrange the letters of 'LEVEL'?
    Ans: $5!\mathrm{/}(2!\times 2!)=30$ (L and E repeat)

15. How many ways to arrange 6 people in a line if 2 must be together?
    Ans: Treat 2 as 1: $5!\times 2!=240$

16. How many ways to arrange 5 books if 2 must not be together?
    Ans: Total: $5!=120$, Together: $4!\times 2=48$, Not together: $120-48=72$

17. How many 3-digit numbers with distinct digits?
    Ans: $9\times 9\times 8=648$ (first digit 1-9, next 9, then 8)

18. How many ways to arrange 7 people in a circle?
    Ans: $(7-1)!=720$

19. How many ways to arrange 4 books if one is always at the end?
    Ans: Fix one at end: $3!\times 2=12$

20. How many ways to arrange 3 red and 2 blue balls in a row?
    Ans: $5!\mathrm{/}(3!2!)=10$

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Combinations (Order Doesn’t Matter)

21. How many ways to select 4 from 10 students?
    Ans: $10C_4=210$

22. How many ways to select a committee of 3 from 7?
    Ans: $7C_3=35$

23. How many ways to select 2 boys and 2 girls from 5 boys and 4 girls?
    Ans: $5C_2\times 4C_2=10\times 6=60$

24. How many ways to select 5 cards from a deck of 52?
    Ans: $52C_5=2,598,960$

25. How many ways to select 2 balls from 5 identical balls?
    Ans: 1 (identical objects)

26. How many ways to select 3 out of 8 if two particular are never together?
    Ans: Total: $8C_3=56$, Together: $7C_2=21$, Not together: $56-21=35$

27. How many ways to select 2 from 6, if order matters?
    Ans: $6P_2=30$

28. How many ways to select 3 out of 6, if order doesn’t matter?
    Ans: $6C_3=20$

29. How many ways to select 3 books from 10?
    Ans: $10C_3=120$

30. How many ways to select 2 pens from 5 different pens?
    Ans: $5C_2=10$

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Probability (Basic)

31. A coin is tossed. What is the probability of getting heads?
    Ans: $\frac{1}{2}$

32. A die is rolled. Probability of getting 4?
    Ans: $\frac{1}{6}$

33. Probability of drawing a king from a deck?
    Ans: $\frac{4}{52}=\frac{1}{13}$

34. Probability of drawing a red card?
    Ans: $\frac{26}{52}=\frac{1}{2}$

35. Probability of getting an even number on a die?
    Ans: $\frac{3}{6}=\frac{1}{2}$

36. Probability of drawing an ace from a deck?
    Ans: $\frac{4}{52}=\frac{1}{13}$

37. Probability of drawing a heart from a deck?
    Ans: $\frac{13}{52}=\frac{1}{4}$

38. Probability of drawing a face card?
    Ans: $\frac{12}{52}=\frac{3}{13}$

39. Probability of getting a number less than 4 on a die?
    Ans: $\frac{3}{6}=\frac{1}{2}$

40. Probability of getting a 2 or 5 on a die?
    Ans: $\frac{2}{6}=\frac{1}{3}$

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Probability with Permutations/Combinations

41. Probability of guessing a 3-digit number with no repeats?
    Ans: $\frac{1}{10P_3}=\frac{1}{720}$2.

42. Probability of selecting 2 red balls from 4 red and 3 blue balls?
    Ans: $\frac{4C_2}{7C_2}=\frac{6}{21}=\frac{2}{7}$

43. Probability of arranging 3 books in a row out of 5?
    Ans: $\frac{5P_3}{5P_3}=1$

44. Probability of drawing 2 aces from a deck?
    Ans: $\frac{4C_2}{52C_2}=\frac{6}{1326}=\frac{1}{221}$

45. Probability of selecting a committee of 2 men and 1 woman from 3 men and 2 women?
    Ans: $\frac{3C_2\times 2C_1}{5C_3}=\frac{3\times 2}{10}=\frac{6}{10}=\frac{3}{5}$

46. Probability of drawing a queen or king from a deck?
    Ans: $\frac{8}{52}=\frac{2}{13}$

47. Probability of getting at least one head in two coin tosses?
    Ans: $1-P(\text{no head})=1-\frac{1}{4}=\frac{3}{4}$

48. Probability of drawing 2 black cards from a deck?
    Ans: $\frac{26C_2}{52C_2}$

49. Probability of drawing 2 cards, both spades?
    Ans: $\frac{13C_2}{52C_2}$

50. Probability of drawing a king and a queen together?
    Ans: $\frac{4\times 4}{52\times 51}$

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Arrangements and Selections (Advanced)

51. How many ways to arrange 5 people if 2 must always be together?
    Ans: $4!\times 2!=48$

52. How many ways to select a group of 3 from 6 men and 4 women, at least 1 woman?
    Ans: $6C_3+6C_2\times 4C_1+6C_1\times 4C_2+4C_3$

53. How many 4-digit numbers can be formed from 1, 2, 3, 4, 5, no repetition?
    Ans: $5P_4=120$

54. How many ways to arrange the letters of 'BANANA'?
    Ans: $6!\mathrm{/}(3!2!)=60$

55. How many ways to select 2 pens from 4 red and 3 blue pens?
    Ans: $7C_2=21$

56. How many ways to select 3 balls from 6, if one particular ball must be included?
    Ans: $5C_2=10$

57. How many ways to arrange 4 boys and 2 girls in a row so that girls are together?
    Ans: $5!\times 2!=240$

58. How many ways to select 2 men and 2 women from 5 men and 4 women?
    Ans: $5C_2\times 4C_2=10\times 6=60$

59. How many ways to select a committee of 3 from 7 people?
    Ans: $7C_3=35$

60. How many ways to arrange 3 vowels from 'EDUCATION'?
    Ans: 5 vowels: $5P_3=60$

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Probability (Mixed Applications)

61. Probability of drawing 2 red cards from a deck?
    Ans: $\frac{26C_2}{52C_2}$

62. Probability of drawing 2 cards, both kings?
    Ans: $\frac{4C_2}{52C_2}$

63. Probability of drawing 2 cards, both of same suit?
    Ans: $\frac{4\times 13C_2}{52C_2}$

64. Probability of drawing 2 cards, both face cards?
    Ans: $\frac{12C_2}{52C_2}$

65. Probability of getting a sum of 7 on two dice?
    Ans: $\frac{6}{36}=\frac{1}{6}$

66. Probability of getting at least one six in two dice throws?
    Ans: $1-(\frac{5}{6}\times \frac{5}{6})=1-\frac{25}{36}=\frac{11}{36}$

67. Probability of getting all heads in 3 coin tosses?
    Ans: $\frac{1}{8}$

68. Probability of getting exactly two heads in 3 coin tosses?
    Ans: $\frac{3}{8}$

69. Probability of getting at least one tail in 3 coin tosses?
    Ans: $1-\frac{1}{8}=\frac{7}{8}$

70. Probability of getting a prime number on a die?
    Ans: $\frac{3}{6}=\frac{1}{2}$

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Counting with Repetition

71. How many ways to select 3 balls from 5 colors with repetition?
    Ans: $(5+3-1)C_3=7C_3=35$

72. How many ways to arrange 3 letters from A, B, C with repetition?
    Ans: $3^3=27$

73. How many 2-digit numbers can be formed from 1, 2, 3 with repetition?
    Ans: $3^2=9$

74. How many ways to select 4 fruits from 6 types with repetition?
    Ans: $9C_4=126$

75. How many ways to arrange 2 letters from 4 with repetition?
    Ans: $4^2=16$

76. How many ways to select 2 pencils from 5 types with repetition?
    Ans: $6C_2=15$

77. How many 4-digit numbers from 0-9 with repetition?
    Ans: ${10}^4=10,000$

78. How many ways to select 3 ice creams from 4 flavors with repetition?
    Ans: $6C_3=20$

79. How many 3-digit numbers from 1-5 with repetition?
    Ans: $5^3=125$

80. How many ways to select 5 balls from 3 colors with repetition?
    Ans: $7C_5=21$

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Probability (Advanced)

81. Probability of getting 2 heads in 4 coin tosses?
    Ans: $\frac{6}{16}=\frac{3}{8}$

82. Probability of drawing 2 balls of different colors from 3 red, 2 blue?
    Ans: $\frac{3C_1\times 2C_1}{5C_2}=\frac{6}{10}=\frac{3}{5}$

83. Probability of getting a sum of 8 on two dice?
    Ans: $\frac{5}{36}$

84. Probability of getting a pair (same number) on two dice?
    Ans: $\frac{6}{36}=\frac{1}{6}$

85. Probability of drawing a spade or a king?
    Ans: $\frac{13+4-1}{52}=\frac{16}{52}=\frac{4}{13}$

86. Probability of drawing 2 cards, at least one ace?
    Ans: $1-\frac{48C_2}{52C_2}$

87. Probability of getting 2 sixes in 2 dice throws?
    Ans: $\frac{1}{36}$

88. Probability of getting at most one head in 3 coin tosses?
    Ans: $\frac{4}{8}=\frac{1}{2}$

89. Probability of drawing 2 cards, both red or both black?
    Ans: $\frac{26C_2+26C_2}{52C_2}$

90. Probability of drawing 2 cards, both not face cards?
    Ans: $\frac{40C_2}{52C_2}$

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Mixed Conceptual and Word Problems

91. How many ways to select a team of 3 from 4 boys and 3 girls?
    Ans: $7C_3=35$

92. How many ways to arrange 3 vowels from 'AEIOU'?
    Ans: $5P_3=60$

93. Probability of drawing a black king from a deck?
    Ans: $\frac{2}{52}=\frac{1}{26}$

94. How many ways to select 2 out of 8, if one particular must be included?
    Ans: $7C_1=7$

95. Probability of drawing 2 queens from a deck?
    Ans: $\frac{4C_2}{52C_2}=\frac{6}{1326}=\frac{1}{221}$

96. How many ways to arrange 4 objects in a row?
    Ans: $4!=24$

97. How many ways to select 2 pens from 5, if order matters?
    Ans: $5P_2=20$

98. Probability of getting a sum of 11 on two dice?
    Ans: $\frac{2}{36}=\frac{1}{18}$

99. Probability of getting a tail in a single coin toss?
    Ans: $\frac{1}{2}$

100. How many ways to arrange the letters of 'SUCCESS'?
     Ans: $7!\mathrm{/}(3!2!)=420$

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Use these notes and questions to master permutations, combinations, and probability for all competitive exams. For more examples and explanations, refer to BYJU’S, Cuemath, and Khan Academy136