Counting, induction and discrete probability: basics of counting, pigeonhole principle, permutations and combinations, inclusion–exclusion principle, mathematical induction, probability, Bayes' theorem. - Question Bank

1. What is the formula for the number of permutations of n objects where there are n1 identical objects of type 1, n2 identical objects of type 2, ..., nk identical objects of type k?
A) n!
B) n! / (n1! + n2! + ... + nk!)
C) n! / (n1! * n2! * ... * nk!)
D) (n - n1 - n2 - ... - nk)!
2. In a group of 20 people, 12 play cricket and 15 play football. If each person plays at least one game, how many play both?
A) 3
B) 5
C) 7
D) 9
3. What is the probability of getting at least one head when flipping a fair coin 4 times?
A) 1/16
B) 7/16
C) 15/16
D) 1
4. How many non-negative integer solutions are there to the equation x1 + x2 + x3 = 10?
A) 120
B) 100
C) 66
D) 45
5. Consider the statement: 1 + 2 + ... + n = n(n+1)/2. If the inductive step assumes P(k) is true, what needs to be proven?
A) 1 + 2 + ... + k = k(k+1)/2
B) 1 + 2 + ... + (k+1) = (k+1)(k+2)/2
C) 1 + 2 + ... + n = n(n+1)/2 for all n
D) The statement is true for n=1
6. If P(A) = 0.6, P(B) = 0.5, P(A U B) = 0.8, what is P(A ∩ B)?
A) 0.1
B) 0.3
C) 0.7
D) 0.9
7. What is the probability of drawing a spade or a heart from a standard deck of 52 cards?
A) 1/4
B) 1/2
C) 13/52
D) 26/52
8. How many ways can the letters of the word 'BOOK' be arranged?
A) 12
B) 24
C) 6
D) 48
9. In the context of the Pigeonhole Principle, what are the 'pigeons' and 'pigeonholes' in the statement: 'If you have 5 pigeons and 4 pigeonholes, at least one pigeonhole must contain more than one pigeon'?
A) Pigeons are the containers, pigeonholes are the items.
B) Pigeons are the items, pigeonholes are the containers.
C) Both are types of containers.
D) Both are types of items.
10. What is the total number of possible outcomes when rolling two distinct dice?
A) 6
B) 12
C) 36
D) 72
11. How many permutations are there of the set {a, b, c}?
A) 3
B) 6
C) 9
D) 27
12. If P(A) = 0.5, P(B) = 0.4, and P(A|B) = 0.8, what is P(A and B)?
A) 0.32
B) 0.4
C) 0.5
D) 0.7
13. What is the probability that a randomly selected integer from 1 to 10 is divisible by 3?
A) 1/10
B) 2/10
C) 3/10
D) 4/10
14. A survey of 100 people found that 70 like coffee and 80 like tea. If 60 people like both, how many like neither?
A) 0
B) 10
C) 20
D) 30
15. If you are proving a statement P(n) for all integers n >= 1, what is the first step in mathematical induction?
A) Prove P(k+1) is true assuming P(k) is true.
B) Prove P(1) is true.
C) Prove P(n) is true for some arbitrary n.
D) Prove P(0) is true.
16. What is the value of P(n, 1)?
A) 0
B) 1
C) n
D) n!
17. What is the value of C(n, n)?
A) 0
B) 1
C) n
D) n!
18. How many ways can you choose a team of 4 players from a group of 7?
A) 7
B) 28
C) 84
D) 210
19. In Bayes' Theorem, P(A|B) = P(B|A) * P(A) / P(B), what is P(A) called?
A) Likelihood
B) Marginal Probability
C) Posterior Probability
D) Prior Probability
20. What is the probability of getting exactly 2 heads in 3 coin flips?
A) 1/8
B) 3/8
C) 4/8
D) 1/2
21. Using the Inclusion-Exclusion Principle for two sets A and B, what is the formula for |A U B|?
A) |A| + |B|
B) |A| - |B|
C) |A| + |B| - |A ∩ B|
D) |A| + |B| + |A ∩ B|
22. Suppose a bag contains 5 red balls and 3 blue balls. If you draw one ball at random, what is the probability it is red?
A) 3/8
B) 5/8
C) 5/3
D) 1/8
23. What is the probability of rolling a sum of 7 with two standard six-sided dice?
A) 1/36
B) 5/36
C) 6/36
D) 7/36
24. How many distinct permutations are there of the letters in the word 'MISSISSIPPI'?
A) 34650
B) 346500
C) 11!
D) 11! / (4! * 4! * 2! * 1!)
25. What is the minimum number of students in a class to guarantee that at least two students receive the same score on a test graded from 0 to 100?
A) 100
B) 101
C) 102
D) 200
26. In how many ways can 5 people be seated in a row?
A) 5
B) 25
C) 120
D) 625
27. If you have 3 shirts and 4 pants, how many different outfits can you create?
A) 7
B) 12
C) 81
D) 64
28. What is the inductive step in a proof by induction?
A) Verifying the statement for the first case
B) Assuming the statement is true for an arbitrary case k and proving it for k+1
C) Showing that the statement holds for all cases
D) Demonstrating the statement is false for some cases
29. What is the base case in a proof by induction for the statement P(n)?
A) Proving P(k) implies P(k+1)
B) Proving P(n) for an arbitrary n
C) Proving P(1) is true
D) Proving P(0) is true
30. If P(A) = 0.6 and P(B) = 0.7 and P(A and B) = 0.4, what is P(A or B)?
A) 0.3
B) 0.9
C) 1.0
D) 1.3
31. What is the probability of drawing an ace from a standard deck of 52 cards?
A) 1/13
B) 1/52
C) 4/13
D) 1/4
32. In a class of 30 students, 15 like Math and 20 like Science. If 5 students like both, how many students like either Math or Science?
A) 25
B) 30
C) 35
D) 40
33. Consider a set of 5 distinct books. How many ways can you arrange 3 of them on a shelf?
A) 10
B) 15
C) 60
D) 125
34. How many different outcomes are possible when flipping a coin 3 times?
A) 3
B) 6
C) 8
D) 9
35. What is the value of 5! (5 factorial)?
A) 5
B) 25
C) 120
D) 600
36. How many ways can the letters of the word 'APPLE' be arranged?
A) 120
B) 60
C) 24
D) 5
37. If events A and B are independent, what is P(A and B)?
A) P(A) + P(B)
B) P(A) * P(B)
C) P(A) / P(B)
D) 1
38. What does Bayes' Theorem describe?
A) The probability of independent events
B) The probability of an event given prior knowledge of related events
C) The sum of probabilities in a sample space
D) The number of permutations
39. What is the formula for conditional probability P(A|B)?
A) P(A and B) / P(B)
B) P(A and B) / P(A)
C) P(A) * P(B)
D) P(A) + P(B)
40. What is the probability of an event that is impossible to occur?
A) 0
B) 0.5
C) 1
D) Undefined
41. What is the probability of an event that is certain to occur?
A) 0
B) 0.5
C) 1
D) Undefined
42. In probability, what is the set of all possible outcomes of an experiment called?
A) Event
B) Sample Space
C) Outcome
D) Probability Distribution
43. What are the two steps involved in a proof by mathematical induction?
A) Base case and Inductive step
B) Assumption and Conclusion
C) Premise and Deduction
D) Hypothesis and Verification
44. What is the principle of mathematical induction used for?
A) Counting permutations
B) Proving statements about natural numbers
C) Calculating probabilities
D) Finding combinations
45. Which principle is used to count the number of elements in the union of multiple sets?
A) Pigeonhole Principle
B) Multiplication Principle
C) Addition Principle
D) Inclusion-Exclusion Principle
46. If a task can be performed in m ways and a second task can be performed in n ways, and these tasks cannot be performed at the same time, then the task can be performed in how many ways?
A) m + n
B) m - n
C) m * n
D) m / n
47. If a task can be performed in m ways, and after it is performed, a second task can be performed in n ways, then the two tasks can be performed in sequence in how many ways?
A) m + n
B) m - n
C) m * n
D) m / n
48. What is the number of combinations of n distinct objects taken r at a time, denoted as C(n, r)?
A) n! / (n-r)!
B) n! / r!
C) (n-r)! / n!
D) n! / (r! * (n-r)!)
49. What is the number of permutations of n distinct objects taken r at a time, denoted as P(n, r)?
A) n! / (n-r)!
B) n! / r!
C) (n-r)! / n!
D) n! / (r! * (n-r)!)
50. In how many ways can a committee of 3 people be chosen from a group of 10 people?
A) 100
B) 120
C) 720
D) 3628800
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