---
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entity_type: "chapter"
id: "66f1595be361cd99fe370dda"
title: "Binomial Theorem"
board: "CBSE"
curriculum: "CBSE"
class: "Class 11"
subject: "Mathematics"
book: "Mathematics"
chapter: "Binomial Theorem"
chapter_slug: "binomial-theorem"
canonical_url: "https://www.edzy.ai/cbse-class-11-mathematics-binomial-theorem"
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source: "Edzy"
version: 1
last_updated: "2026-06-20"
---

# Binomial Theorem
Mathematics is a most exact science and its conclusions are capable of absolute proofs. This chapter explores the binomial theorem, a technique essential for expanding expressions of the form $(a + b)^{n}$ for positive integral indices.

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## Knowledge Snapshot

| Field | Details |
| :--- | :--- |
| Class | Class 11 |
| Subject | Mathematics |
| Book | Mathematics |
| Chapter | Binomial Theorem |
| Pages | 126-134 |

---

## Chapter Summary

### Short Summary
The chapter introduces the binomial theorem for positive integral indices, facilitating easier expansions of binomials and providing formulas for their coefficients.

### Detailed Summary
The chapter describes the method to expand $(a + b)^{n}$ using the binomial theorem, demonstrating how to derive coefficients from Pascal's triangle. It includes identities, examples of expansions, special cases, and proofs of the theorem using mathematical induction. The coefficients in these expansions represent the binomial coefficients arranged in Pascal's triangle. The chapter concludes with numerous exercises to apply concepts learned.

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## Topic-Wise Explanation

### Introduction
The introduction emphasizes the historical context of learning binomial expansions and highlights the need for the binomial theorem in simplifying higher power calculations.

### Binomial Theorem for Positive Integral Indices
The theorem states that $(a + b)^{n} = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k}$. It is derived and proven using mathematical induction.

### Pascal's Triangle
Pascal's triangle is depicted as a triangular array where each number is the sum of the two directly above it, facilitating the extraction of coefficients for binomial expansions.

### Observations on Binomial Expansion
This section lists key observations regarding the number of terms, powers of variables, and the roles of coefficients in binomial expansions.

### Examples of Binomial Expansion
Examples demonstrate how to apply the binomial theorem through expansions such as $(2x + 3y)^{5}$ and others, showcasing the ease of calculations using the theorem.

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## Core Ideas

| Idea | Explanation |
| :--- | :--- |
| Binomial Expansion | Method of expressing $(a + b)^{n}$ using binomial coefficients and powers of $a$ and $b$. |

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## Key Concepts

| Concept | Meaning |
| :--- | :--- |
| Binomial Theorem | A formula for the expansion of powers of binomials. |
| Binomial Coefficient | The coefficients in the binomial expansion, denoted as ${n \choose k}$. |

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## Important Points for Revision

* The expansion of $(a + b)^{n}$ produces $(n + 1)$ terms.
* Coefficients are derived from Pascal's triangle or calculated using the formula ${n \choose k}$.
* Each term's indices of $a$ and $b$ must sum to $n$.
* The theorem applies only for positive integral indices in this chapter.
* Special cases include substitutions like $(x - y)^{n}$ and $(1 + x)^{n}$.
* The proof of the theorem utilizes induction.
* Coefficients can be rewritten in terms of combinations to simplify calculations.
* Extensive exercises are provided for practice.

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## Practice Questions

### Short Answer Questions
1. Expand $(1 - 2x)^{5}$.
2. Compute $(98)^{5}$ using the binomial theorem.
3. What is the expansion of $(x + 2)^{6}$?
4. How many terms does $(a + b)^{7}$ have?
5. Prove that $6^{n} - 5^{n}$ leaves a remainder of 1 when divided by 25.

### Long Answer Questions
1. Prove the Binomial Theorem by induction.
2. Expand $(4 + 2x)^{3}$ and find the coefficients.
3. Demonstrate the relationship between the binomial coefficients and their arrangement in Pascal's triangle.

---

## Source Attribution

| Field | Value |
| :--- | :--- |
| Source | Edzy |
| Reference Type | examSubjectBookChapter |
| Reference ID | 66f1595be361cd99fe370dda |
| Canonical URL | https://www.edzy.ai/cbse-class-11-mathematics-binomial-theorem |
| Markdown URL | https://www.edzy.ai/okf/chapter/cbse-class-11-mathematics-binomial-theorem.md |