🔢 Quant

Modern Maths

Permutation & Combination, Probability, Set Theory, Progressions. Usually 3–5 questions.

of Quant

Why This Topic Matters

Total PYQs📊

of 1002 · 2021–2025

Years featured📅

5/5

of recent CAT years

% of Quant📈

~5%

of section questions

Est. hours⏱️

~8h

to master

1/22

2021

~2/22

2022

~1/22

2023

~2/22

2024

~2/22

2025

Few questions, high yield

Modern Maths — Permutations & Combinations, Probability, and Set Theory — is roughly 1 of the 22 Quant questions per slot. The concepts are small in number and reusable, so the return on mastering them is high. Each area has its own page; here are the load-bearing ideas.

What CAT 2021–2025 actually asked

Sub-skill	2021	2022	2023	2024	2025	Avg/slot
Permutation & Combination	1.0	1.0	0.3	1.0	1.0	0.9
Set theory (often with statistics)	–	0.3	0.3	–	0.3	0.2
Probability	–	–	–	0.3	–	0.1

🎯PYQ Evidence

Budget by the data: P&C first, the rest second. Counting questions appeared every year (13 of the 17 Modern Math questions). Pure probability appeared exactly once in five years (2024) — and even that one leaned on counting. Set theory surfaced three times, usually fused with averages/statistics. If you have limited hours, drill P&C deeply and treat probability as "P&C divided by a total," which is genuinely what CAT makes it.

Counting — permutations vs combinations

Fundamental rule: if a task splits into stages with $m$ and $n$ choices, the total is $m\times n$ (AND ⇒ multiply); mutually exclusive cases add (OR ⇒ add).
Order matters → permutations: $^nP_r=\dfrac{n!}{(n-r)!}$ .
Order doesn't matter → combinations: $^nC_r=\dfrac{n!}{r!\,(n-r)!}$ , with the symmetry $^nC_r={}^nC_{\,n-r}$ .

⚠️CAT Trap

The order-matters slip. The single most common Modern Math error is using a permutation where a combination is meant (or vice-versa). Ask: would swapping two chosen items create a genuinely different outcome? If no, it's a combination. Two siblings of this trap that CAT loves: seating around a circle divides a row count by the rotations ( $n! \to (n-1)!$ ), and identical objects divide by the repeats ( $\frac{n!}{a!\,b!}$ ). Both appeared in recent papers' counting questions — the wrong option is always the un-divided count.

Probability — and the complement trick

Basic probability is $\dfrac{\text{favourable}}{\text{total}}$ , but on CAT the fast route is often the complement:

$P(\text{at least one})=1-P(\text{none})$

Build a probability tree when events happen in stages: multiply along a path (AND), add across disjoint paths (OR), and the branches at any node sum to 1. Conditional probability is

$P(A\mid B)=\frac{P(A\cap B)}{P(B)}.$

Set theory — the inclusion–exclusion formulas

Two sets: $\;|A\cup B|=|A|+|B|-|A\cap B|$ .

Three sets:

$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|B\cap C|-|C\cap A|+|A\cap B\cap C|$

A Venn diagram with the innermost region filled first (working outward) handles almost every CAT set question.

A worked example

✏️Worked Example

A bag holds 4 red and 6 blue balls. Two are drawn at random. What is the probability of at least one red?

Go through the complement — "no red" means both blue:

$P(\text{both blue})=\frac{6}{10}\times\frac{5}{9}=\frac{30}{90}=\frac13.$

$P(\text{at least one red})=1-\frac13=\boxed{\dfrac23}.$

Counting the "at least one" cases directly would mean adding exactly one red and both red — the complement does it in one line.

Where to go deeper

Dedicated pages: Permutations & Combinations, Probability, Set Theory (and Sequences & Series, housed under Algebra).

🎯PYQ Evidence

Counting problems split into three moves: multiply independent choices, subtract bad cases, or squeeze an overlap. : the mandatory picks multiply (5×4×2 = 40) while the optional sauces (0, 1, or 2 of 6) are counted as C(6,0)+C(6,1)+C(6,2) = 22 and added, giving 880. : count all 3⁶ functions, then inclusion-exclusion subtracts those missing a target — C(3,1)·2⁶ back to C(3,2)·1⁶ — leaving 540 surjections. : the all-three count is squeezed between the smallest single group (52) above and the totals-minus-class-size forced minimum below, and the gap is the answer. Ask whether the choices are independent, whether to remove forbidden cases, or whether you're bounding an overlap.

Checklist

Decide permutation vs combination by asking if order matters
Circle seating → $(n-1)!$ ; identical objects → divide by repeats
Reach for $1-P(\text{none})$ on "at least one" questions
Multiply along a tree path (AND), add across paths (OR)
Fill the innermost Venn region first
Keep the 3-set inclusion–exclusion formula at your fingertips

Sample Questions

36 practice questions

TITAMedium

The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c, and d, with each digit appearing exactly once in every number, is 173307 + n, where n is a single digit natural number. Then, the value of (a + b + c + d + n) is

Your answer

Easy

The arithmetic mean of all the distinct numbers that can be obtained by rearranging the digits in 1182, including itself, is

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CAT PYQ Spotlight

Actual CAT questions on this topic

CAT 2025 · Slot 1

TITAMedium

The number of distinct pairs of integers (x, y) satisfying the inequalities x > y ≥ 3 and x + y < 14 is

Your answer

CAT 2024 · Slot 1

TITAHard

The sum of all four-digit numbers that can be formed with the distinct non-zero digits a, b, c, and d, with each digit appearing exactly once in every number, is 153310 + n, where n is a single digit natural number. Then, the value of (a + b + c + d + n) is

Your answer

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