IIM CAT 2026 cheat sheet for Modern Math! 📱 Practice 1 Lakh+ PYQs on ExamBhai.com! - Free Study Material

Chapter Summary: Modern Math for CAT 2026

Modern Mathematics is a high-yield section in the Quantitative Aptitude module of the IIM CAT exam. It tests logical deduction, counting principles, and foundational abstract concepts. Below is a structured breakdown of core formulas, properties, and analytical frameworks.

1. Permutations and Combinations

The fundamental principles of counting lay the groundwork for solving selection and arrangement problems.

• Fundamental Principle of Counting: If task \(A\) occurs in \(m\) ways and task \(B\) occurs in \(n\) ways, both occur sequentially in \(m \times n\) ways (Multiplication Rule). If either occurs, it takes \(m + n\) ways (Addition Rule).
• Permutations (Arrangements): The total ways to arrange \(r\) objects out of \(n\) distinct items: \[P(n, r) = \frac{n!}{(n - r)!}\]
• Combinations (Selections): The total ways to select \(r\) items out of \(n\) distinct items: \[C(n, r) = \frac{n!}{r!(n - r)!}\]
• Circular Permutations: Arrangement of \(n\) distinct items around a circle is \((n - 1)!\). If clockwise and counter-clockwise orders are indistinguishable, it reduces to \(\frac{(n - 1)!}{2}\).

2. Probability and Expectation

Probability evaluates the likelihood of target outcomes relative to the entire sample space.

• Classical Probability: \(P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Sample Space}}\), where \(0 \le P(A) \le 1\).
• Conditional Probability: Probability of event \(A\) occurring given that event \(B\) has occurred: \[P(A | B) = \frac{P(A \cap B)}{P(B)}\]
• Independent Events: Events \(A\) and \(B\) satisfy \(P(A \cap B) = P(A) \times P(B)\).

3. Set Theory and Venn Diagrams

Set theory deals with grouped entities and visual overlaps using Euler-Venn diagrams.

• Principle of Inclusion-Exclusion (2 Sets): \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\)
• Principle of Inclusion-Exclusion (3 Sets): \[n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)\]

4. Sequences, Series, and Progressions

• Arithmetic Progression (AP): \(n\)-th term \(T_n = a + (n - 1)d\); Sum \(S_n = \frac{n}{2} \bigl(2a + (n - 1)d\bigr)\).
• Geometric Progression (GP): \(n\)-th term \(T_n = a r^{n - 1}\); Sum of infinite terms (for \(|r| < 1\)) is \(S_{\infty} = \frac{a}{1 - r}\).
• AM-GM-HM Inequality: For positive real numbers: \[\text{AM} \ge \text{GM} \ge \text{HM}\]