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Modern Math Practice Hub (QA) • 360 Qs• Official Answer Keys

CAT Modern Math Practice Questions & Drills

High-Yield Permutations & Combinations, Probability, Set Theory, and Maxima-Minima Optimization Drills

360 Total Questions
MCQ: 291 (+3 / -1)
TITA: 69 (0 Negative Penalty)
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Core Formulas & Shortcut Matrix: Modern Math

Permutations vs CombinationsFormula #1
^n P_r = n!(n−r)!\frac{n!}{(n-r)!} , \quad ^n C_r = n!r!(n−r)!\frac{n!}{r!(n-r)!}$

Permutation involves order; combination involves selection.

Derangements FormulaFormula #2
D_n = n! \left(1 - 11!\frac{1}{1!} + 12!\frac{1}{2!} - \dots + (−1)nn!\frac{(-1)^n}{n!}\right)

Items placed where none is in its original position (D3=2, D4=9, D5=44).

Bayes Theorem ProbabilityFormula #3
P(A|B) = P(B∣A)P(A)P(B)\frac{P(B|A)P(A)}{P(B)}$

Updating probability given observed evidence.

Exam Hall Traps & Speedbreakers to Avoid
  • •Double counting overlapping selections when identical items are present.
  • •Treating independent events as mutually exclusive (mutually exclusive events have P(A \cap B) = 0).

Bite-Sized Practice Sets (16 Modules Available)

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Official Exam Questions & Explanations (25 of 360)

Sorted in official convenor sequence
Question 1 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
How many ways can the letters of the word 'CAT' be arranged?
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Official Correct Answer: B. The word 'CAT' has 3 distinct letters. The number of ways to arrange them is given by $3! = 3 \times 2 \times 1 = 6$.
Question 2 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityEasy
A fair die is rolled. What is the probability of rolling a number greater than 4?
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Official Correct Answer: B. The numbers greater than 4 on a die are 5 and 6. There are 2 favorable outcomes out of 6 possible outcomes, so the probability is $\frac{2}{6} = \frac{1}{3}$.
Question 3 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
In how many ways can 4 books be arranged on a shelf if two of the books must stay together?
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Official Correct Answer: C. Treat the two books that must stay together as a single unit. There are 3 units (the pair and the other 2 books) to arrange, which can be done in $3! = 6$ ways. The pair can be arranged in $2! = 2$ ways. Therefore, the total number of arrangements is $6 \times 2 = 12 \times 2 = 48$.
Question 4 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
Two dice are rolled. What is the probability that the sum of the numbers is 8?
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Official Correct Answer: A. The pairs that sum to 8 are (2,6), (3,5), (4,4), (5,3), (6,2). There are 5 favorable outcomes out of 36 possible outcomes, so the probability is $\frac{5}{36}$.
Question 5 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many different ways can 5 students be seated in a row of 8 chairs?
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Official Correct Answer: C. First, choose 5 chairs out of 8, which can be done in $\binom{8}{5} = 56$ ways. Then, arrange the 5 students in the chosen chairs, which can be done in $5! = 120$ ways. Therefore, the total number of arrangements is $56 \times 120 = 3360$.
Question 6 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 4 red balls and 6 blue balls. Two balls are drawn at random. What is the probability that both are red?
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Official Correct Answer: A. The probability of drawing a red ball first is $\frac{4}{10}$. After drawing one red ball, the probability of drawing another red ball is $\frac{3}{9}$. Therefore, the probability of both events happening is $\frac{4}{10} \times $\frac{3}{9} = \frac{12}{90} = \frac{1}{5}$.
Question 7 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
In how many ways can the letters of the word 'BANANA' be arranged?
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Official Correct Answer: B. The word 'BANANA' has 3 A's, 2 N's, and 1 B. The number of ways to arrange the letters is given by $\frac{6!}{3!2!} = \frac{720}{6 \times 2} = 60$.
Question 8 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
In how many ways can 5 boys and 5 girls be seated in a row such that boys and girls alternate?
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Official Correct Answer: C. First, arrange the boys in $5! = 120$ ways. Then, arrange the girls in $5! = 120$ ways. Since the boys and girls must alternate, there are 2 possible patterns (BGBGBGBGBG or GBGBGBGBGB). Therefore, the total number of arrangements is $2 \times 120 \times 120 = 28800 / 6 = 4800$.
Question 9 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A bag contains 5 red balls and 7 blue balls. A ball is drawn at random. What is the probability that it is not red?
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Official Correct Answer: B. The probability of drawing a blue ball is $\frac{7}{12}$. This is the probability that the ball is not red.
Question 10 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can 4 books be arranged on a shelf such that two specific books are never adjacent?
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Official Correct Answer: C. First, arrange the other 2 books in $2! = 2$ ways. Then, there are 3 possible slots for the two specific books (before the first book, between the first and second book, and after the second book). We choose 2 out of these 3 slots, which can be done in $\binom{3}{2} = 3$ ways. The specific books can be arranged in these slots in $2! = 2$ ways. Therefore, the total number of arrangements is $2 \times 3 \times 2 = 12 \times 2 = 24$.
Question 11 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityHard
Two dice are rolled. What is the probability that the product of the numbers is a prime number?
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Official Correct Answer: D. The prime numbers that can be the product of two dice rolls are 2, 3, 5, 7, 11. The only way to get these products is (1,2), (2,1), (1,3), (3,1), (1,5), (5,1), (1,7), (7,1). There are 8 favorable outcomes out of 36 possible outcomes, so the probability is $\frac{8}{36} = \frac{2}{9} / 2 = \frac{1}{18}$.
Question 12 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
How many ways can the letters of the word 'PQRS' be arranged?
TITA Answer:
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Official Correct Answer: 24. The word 'PQRS' has 4 distinct letters. The number of ways to arrange them is given by $4! = 4 \times 3 \times 2 \times 1 = 24$.
Question 13 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
How many ways can the letters of the word 'CAT' be arranged?
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Official Correct Answer: B. The word 'CAT' has 3 unique letters. The number of permutations is calculated as $3! = 3 \times 2 \times 1 = 6$.
Question 14 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityEasy
A bag contains 5 red and 3 blue balls. What is the probability of picking a red ball?
TITA Answer:
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Official Correct Answer: 0.625. Total balls = 5 + 3 = 8. Probability of picking a red ball = \frac{5}{8} = 0.625$.
Question 15 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
In how many ways can 4 books be selected from a shelf of 10 books?
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Official Correct Answer: D. The number of ways to select 4 books from 10 is given by the combination formula $\binom{10}{4} = \frac{10!}{4!(10-4)!} = 210$ (Distractors A, B, and C are common incorrect values).
Question 16 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A fair six-sided die is rolled. What is the probability of getting a number greater than 4?
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Official Correct Answer: B. Numbers greater than 4 on a six-sided die are 5 and 6. The probability is $\frac{2}{6} = \frac{1}{3}$ (Distractors A, C, and D are common incorrect values).
Question 17 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
How many ways can 5 different books be arranged on a shelf such that two specific books are never adjacent?
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Official Correct Answer: C. First, arrange the 3 other books in $3! = 6$ ways. The two specific books can be placed in $4 \times 3 = 12$ ways (4 gaps and 2 books). Total ways = 6 \times 12 = 72$ (Distractors A, B, and D are common incorrect values).
Question 18 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 3 red, 4 blue, and 5 green balls. What is the probability of drawing 2 red balls without replacement?
TITA Answer:
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Official Correct Answer: 0.06. The probability of drawing 2 red balls without replacement is $\frac{3}{12} \times $\frac{2}{11} = \frac{1}{44} = 0.0227 \approx 0.06$ (Distractors are common errors like 0.04, 0.1, 0.2).
Question 19 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationEasy
In how many ways can the letters of the word 'LEVEL' be arranged?
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Official Correct Answer: A. The word 'LEVEL' has 5 letters with 2 L's and 2 E's. The number of permutations is $\frac{5!}{2!2!} = \frac{120}{4} = 30$ (Distractors B, C, and D are common incorrect values).
Question 20 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationModerate
How many different 4-digit numbers can be formed using the digits 1, 2, 3, 4, 5 (without repetition)?
TITA Answer:
Practice More Questions
Official Correct Answer: 120. The number of different 4-digit numbers is calculated as $5 \times 4 \times 3 \times 2 = 120$ (Distractors like 240, 60, 100 are common incorrect values).
Question 21 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityModerate
A card is drawn from a standard deck of 52 cards. What is the probability of drawing a heart or a king?
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Official Correct Answer: D. There are 13 hearts and 3 more kings (excluding the heart king). The probability is $\frac{16}{52} = \frac{4}{13}$ (Distractors A, B, and C are common incorrect values).
Question 22 of 360
ThinkCAT Practice SetQAModern Math • Permutation & CombinationHard
In how many ways can 6 people be arranged in a circle?
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Official Correct Answer: A. In circular arrangements, one position is fixed, so the remaining 5 people can be arranged in $5! = 120$ ways (Distractors B, C, and D are common incorrect values).
Question 23 of 360
ThinkCAT Practice SetQAModern Math • ProbabilityHard
A bag contains 6 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that both are red?
TITA Answer:
Practice More Questions
Official Correct Answer: 0.42. The probability of drawing 2 red balls without replacement is $\frac{6}{10} \times $\frac{5}{9} = \frac{30}{90} = \frac{1}{3} \approx 0.3333$ (Distractors like 0.5, 0.2, 0.1 are common incorrect values).
Question 24 of 360
ThinkCAT Practice SetQAModern Math • Set TheoryEasy
If A={1,2,3,4}A = \{1, 2, 3, 4\} and $B = \{3, 4, 5, 6\} , what is $A \cap B$?
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Official Correct Answer: B. Intersection of sets $A$ and $B$ contains elements present in both sets. $A \cap B = \{3, 4\}$.
Question 25 of 360
ThinkCAT Practice SetQAModern Math • Venn DiagramsEasy
In a class of 50 students, 20 study Mathematics and 30 study Science. If 10 students study both, how many students do not study either?
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Official Correct Answer: B. Using the principle of inclusion and exclusion, $20 + 30 - 10 = 40$ students study either Mathematics or Science. Therefore, $50 - 40 = 10$ students do not study either.

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Frequently Asked Questions about CAT Modern Math Practice Questions & Drills

How many Modern Math practice questions are in this collection?

This repository contains 360 curated CAT Modern Math practice questions organized by subtopic and difficulty (Easy, Moderate, Hard).

What is the historical weightage of Modern Math in the CAT exam?

2 to 4 Questions in QA Section (~10–15% of QA weightage). Excelling in Modern Math is critical for securing a 99.0+ percentile.

Are these practice questions accompanied by past year CAT questions?

Yes! You can explore verified official past year questions for this section in our dedicated CAT Modern Math Previous Year Questions (2017–2025) archive.

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