Class 11 Maths Chapter 7 Binomial Theorem NCERT Solutions | Complete Guide with Examples

Introduction

Binomial Theorem is a powerful tool that helps us expand expressions like (a + b)ⁿ without multiplying repeatedly.

This chapter is very useful in algebra, probability, and higher mathematics. Once you understand the pattern, solving questions becomes fast and logical.

Key Formula (Binomial Expansion)

$$(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$$

 For (a − b)ⁿ

(a − b)ⁿ = nC0 aⁿ − nC1 aⁿ⁻¹b + nC2 aⁿ⁻²b² − ...

Important Terms

• General term:
  Tₖ₊₁ = nCk aⁿ⁻ᵏ bᵏ

• Middle term depends on n (odd or even)

• Binomial coefficients: nCk

NCERT SOLVED EXAMPLES (STEP BY STEP)

 Example 1

Expand (1 − 2x)⁵

Solution:
Using binomial expansion:

= 5C0(1)⁵ − 5C1(1)⁴(2x) + 5C2(1)³(2x)² − 5C3(1)²(2x)³ + 5C4(1)(2x)⁴ − 5C5(2x)⁵

= 1 − 10x + 40x² − 80x³ + 80x⁴ − 32x⁵

Example 2

Expand (2/x − x/2)⁵

Solution:
Apply binomial formula carefully:

= 32/x⁵ − 40/x³ + 20/x − 5x + (5x³)/8 − x⁵/32

Example 3

Expand (2x − 3)⁶

Solution:

= 64x⁶ − 576x⁵ + 2160x⁴ − 4320x³ + 4860x² − 2916x + 729

 Example 4

Expand (x/3 + 1/x)⁵

Solution:

= x⁵/243 + 5x³/81 + 10x/27 + 10/(9x) + 5/(3x³) + 1/x⁵

Example 5

Find (a + b)⁴ − (a − b)⁴

Solution:

Expand both and subtract:

= 8ab(a² + b²)

 Example 6

Evaluate (√3 + √2)⁴ − (√3 − √2)⁴

Solution:

= 40√6

ALL EXERCISE TYPES WITH SOLUTIONS

NCERT Exercise 7.1 mainly focuses on expansion problems.

 Type 1: Simple Expansion

(1 − x)⁴

= 1 − 4x + 6x² − 4x³ + x⁴

Type 2: Variable Expansion

(x + 2)³

= x³ + 6x² + 12x + 8

 Type 3: Fraction Type

(1/x + x)³

Solve term by term carefully

 Type 4: Power Expansion

(a + b)⁵

= a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Type 5: Identity Based

(a + b)⁴ − (a − b)⁴

= 8ab(a² + b²)

15 FAQs

1. What is Binomial Theorem?

It is a formula to expand (a + b)^n.

2. What is nCk?

It is binomial coefficient used in expansion.

3. What is general term?

T(k+1) = nCk a^(n-k) b^k

4. What is middle term?

Depends on whether n is odd or even.

5. What is expansion of (a+b)^2?

a^2 + 2ab + b^2

6. What is expansion of (a+b)^3?

a^3 + 3a^2b + 3ab^2 + b^3

7. What happens in (a-b)^n?

Signs alternate.

8. What is Pascal Triangle?

It gives binomial coefficients.

9. What is first term?

a^n

10. What is last term?

b^n

11. What is coefficient?

Numerical factor of a term.

12. Why is binomial theorem important?

Used in algebra and probability.

13. What is value of 0C0?

1

14. What is symmetry property?

nCr = nC(n-r)

15. Where is it used?

In higher maths and competitive exams.