UGC NET - Counting, induction and discrete probability: basics of counting, pigeonhole principle, permutations and combinations, inclusion–exclusion principle, mathematical induction, probability, Bayes' theorem. Online Test (MCQ Quiz)

1. What is the fundamental principle that states if n items are put into m containers, with n > m, then at least one container must contain more than one item?

A) Pigeonhole Principle B) Inclusion-Exclusion Principle C) Multiplication Principle D) Addition Principle

2. 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

3. 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)!)

4. 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)!)

5. 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

6. 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

7. 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

8. What is the principle of mathematical induction used for?

A) Counting permutations B) Proving statements about natural numbers C) Calculating probabilities D) Finding combinations

9. 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

10. In probability, what is the set of all possible outcomes of an experiment called?

A) Event B) Sample Space C) Outcome D) Probability Distribution

11. What is the probability of an event that is certain to occur?

A) 0 B) 0.5 C) 1 D) Undefined

12. What is the probability of an event that is impossible to occur?

A) 0 B) 0.5 C) 1 D) Undefined

13. 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)

14. 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

15. 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

16. How many ways can the letters of the word 'APPLE' be arranged?

A) 120 B) 60 C) 24 D) 5

17. What is the value of 5! (5 factorial)?

A) 5 B) 25 C) 120 D) 600

18. How many different outcomes are possible when flipping a coin 3 times?

A) 3 B) 6 C) 8 D) 9

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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

25. If you have 3 shirts and 4 pants, how many different outfits can you create?

A) 7 B) 12 C) 81 D) 64

26. In how many ways can 5 people be seated in a row?

A) 5 B) 25 C) 120 D) 625

27. 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

28. 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!)

29. 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

30. 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

31. 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|

32. 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

33. 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

34. How many ways can you choose a team of 4 players from a group of 7?

A) 7 B) 28 C) 84 D) 210

35. What is the value of C(n, n)?

A) 0 B) 1 C) n D) n!

36. What is the value of P(n, 1)?

A) 0 B) 1 C) n D) n!

37. 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.

38. 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

39. 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

40. 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

41. How many permutations are there of the set {a, b, c}?

A) 3 B) 6 C) 9 D) 27

42. What is the total number of possible outcomes when rolling two distinct dice?

A) 6 B) 12 C) 36 D) 72

43. 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.

44. How many ways can the letters of the word 'BOOK' be arranged?

A) 12 B) 24 C) 6 D) 48

45. 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

46. 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

47. 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

48. How many non-negative integer solutions are there to the equation x1 + x2 + x3 = 10?

A) 120 B) 100 C) 66 D) 45

49. 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

50. 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

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

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