GRE Quantitative Reasoning Practice: Counting, Probability and Sets
- Counting, Probability and Sets
- 10 questions
- 15 minutes
These questions test the counting principle, permutations and combinations, overlapping sets and basic probability. Decide first whether order matters and whether an item can be used twice; most mistakes come from that choice, not from the arithmetic.
Tap an option to mark your answer. You can change your mind, or tap the same option again to leave the question blank. The correct answers and explanations appear when you finish the test.Tap an option to answer; tap it again to leave the question blank.
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Question 1
How many different arrangements of the letters in the word LEVEL are possible?
Correct answer: C
Five letters can be arranged in 5! = 120 ways, but the two L's and the two E's are identical, so divide by 2! × 2!: 120 ÷ 4 = 30. Dividing by only one 2! gives 60.
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Question 2
In how many ways can a committee of 3 people be chosen from a group of 8 people?
Correct answer: B
The order of committee members does not matter, so the count is 8 × 7 × 6 ÷ (3 × 2 × 1) = 56. The value 336 counts ordered choices, such as a president, a secretary and a treasurer.
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Question 3
Two fair six-sided dice are rolled.
What is the probability that the sum of the numbers rolled is 8?
Correct answer: A
The pairs (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2) give 8, which is 5 of the 36 equally likely outcomes. A sum of 7, with 6 outcomes, would have probability 16.
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Question 4
In a group of 120 students, 70 study French, 50 study Spanish, and 20 study both languages.
How many of the students study neither language?
Correct answer: D
Students studying at least one language: 70 + 50 − 20 = 100. That leaves 120 − 100 = 20 who study neither. Forgetting the overlap gives 120 − 120 = 0.
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Question 5
A fair coin is tossed 3 times.
What is the probability of getting at least one head?
Correct answer: E
The only outcome without a head is three tails, with probability (12)³ = 18. So the probability of at least one head is 1 − 18 = 78.
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Question 6
A committee of 4 is to be chosen from 5 men and 4 women.
How many different committees contain exactly 2 women?
Correct answer: C
Choose 2 of the 5 men in 10 ways and 2 of the 4 women in 6 ways, so there are 10 × 6 = 60 committees. The value 126 counts all committees of 4 from 9 people, with no condition.
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Question 7
A bag holds 4 red marbles and 6 blue marbles. Two marbles are drawn at random, one after the other, without replacement.
What is the probability that both marbles are red?
Correct answer: A
The first draw is red with probability 410, and then 3 of the remaining 9 marbles are red, so the probability is 410 × 39 = 1290 = 215. The value 425 would be correct only with replacement.
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Question 8
Five people are to sit in a row of five chairs. Two of them, Ana and Ben, insist on sitting next to each other.
How many seating arrangements are possible?
Correct answer: B
Treat Ana and Ben as one block: the block and the other 3 people can be arranged in 4! = 24 ways, and inside the block they can sit in 2 orders, so 24 × 2 = 48. Forgetting the 2 orders gives 24.
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Question 9
How many three-digit positive integers have only odd digits?
Correct answer: E
Each of the three places can be 1, 3, 5, 7 or 9, so there are 5 × 5 × 5 = 125 such integers. Zero is even, so the leading-zero problem does not arise here.
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Question 10
Events A and B are independent. The probability of A is 0.4 and the probability of B is 0.5.
What is the probability that A or B (or both) occurs?
Correct answer: D
For independent events the probability of both is 0.4 × 0.5 = 0.2, so P(A or B) = 0.4 + 0.5 − 0.2 = 0.7. Adding the probabilities to get 0.9 counts the overlap twice.
Questions you leave blank are not counted as wrong.
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