Algebra Foundation Reading Time: 15 min JEE Weightage: 4-6%

Binomial Theorem: Introduction & The First Principle

From basic expansion patterns to Pascal's Triangle - build your foundation for JEE success.

4-6%

JEE Weightage

100%

Appears Every Year

Key Concepts

1665

Year Discovered

What is the Binomial Theorem?

The Binomial Theorem provides a powerful formula to expand expressions of the form $(a + b)^n$ where $n$ is a positive integer. Instead of multiplying $(a + b)$ by itself $n$ times, we can use this theorem to directly write the expansion.

📜 Historical Context

The Binomial Theorem was discovered by Sir Isaac Newton in 1665. While the case for positive integer exponents was known to Indian and Persian mathematicians, Newton generalized it to rational exponents, opening doors to calculus and infinite series.

Why Learn Binomial Theorem for JEE?

Direct Questions: 1-2 questions appear every year in JEE Main
Tool for Other Topics: Essential for probability, sequences, and calculus
Time Saver: Avoid lengthy multiplication in problems
Conceptual Foundation: Builds intuition for series and patterns

🎯 Quick Navigation

First Principle Pascal's Triangle Binomial Formula Practice

1. Understanding Through First Principle

Let's build intuition by expanding binomials manually for small values of $n$:

Manual Expansion Patterns

For n = 0:

$$(a + b)^0 = 1$$

For n = 1:

$$(a + b)^1 = a + b$$

For n = 2:

$$(a + b)^2 = (a + b)(a + b) = a^2 + 2ab + b^2$$

For n = 3:

$$(a + b)^3 = (a + b)(a + b)(a + b) = a^3 + 3a^2b + 3ab^2 + b^3$$

For n = 4:

$$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$$

🔍 Key Observations

Number of terms: $n + 1$ terms in the expansion
Powers of a: Decrease from $n$ to $0$
Powers of b: Increase from $0$ to $n$
Sum of powers: In each term, power of a + power of b = $n$
Coefficients: Follow a symmetric pattern

2. Pascal's Triangle - The Coefficient Pattern

Blaise Pascal discovered this beautiful triangular arrangement of binomial coefficients that gives us the coefficients for $(a + b)^n$ expansion.

Pascal's Triangle

n = 0

n = 1

1 1

n = 2

1 2 1

n = 3

1 3 3 1

n = 4

1 4 6 4 1

n = 5

1 5 10 10 5 1

Pattern: Each number is the sum of the two numbers directly above it.

🎯 How to Use Pascal's Triangle

Example: Find expansion of $(x + y)^4$

Step 1: Look at row for n = 4: 1, 4, 6, 4, 1

Step 2: Write terms with decreasing powers of x and increasing powers of y:

$1\cdot x^4y^0 + 4\cdot x^3y^1 + 6\cdot x^2y^2 + 4\cdot x^1y^3 + 1\cdot x^0y^4$

Step 3: Simplify:

$x^4 + 4x^3y + 6x^2y^2 + 4xy^3 + y^4$

3. The Binomial Theorem Formula

For any positive integer $n$, the binomial theorem states:

Binomial Theorem Formula

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

where $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Written in Expanded Form

$$(a + b)^n = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + \cdots + \binom{n}{n}b^n$$

📚 Understanding the Notation

$\binom{n}{r}$ - Binomial Coefficient

Read as "n choose r"

Formula: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$

Also written as: $^nC_r$ or $C(n, r)$

Properties:

$\binom{n}{0} = \binom{n}{n} = 1$
$\binom{n}{r} = \binom{n}{n-r}$ (Symmetric)
$\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$

Example: Expand $(2x + 3)^4$

Step 1: Identify a = 2x, b = 3, n = 4

Step 2: Use binomial theorem:

$(2x + 3)^4 = \binom{4}{0}(2x)^4 + \binom{4}{1}(2x)^3(3) + \binom{4}{2}(2x)^2(3)^2 + \binom{4}{3}(2x)(3)^3 + \binom{4}{4}(3)^4$

Step 3: Calculate coefficients:

$\binom{4}{0} = 1$, $\binom{4}{1} = 4$, $\binom{4}{2} = 6$, $\binom{4}{3} = 4$, $\binom{4}{4} = 1$

Step 4: Compute each term:

$1\cdot 16x^4 + 4\cdot 8x^3\cdot 3 + 6\cdot 4x^2\cdot 9 + 4\cdot 2x\cdot 27 + 1\cdot 81$

Step 5: Simplify:

$16x^4 + 96x^3 + 216x^2 + 216x + 81$

4. The General Term

In the expansion of $(a + b)^n$, the $(r+1)^{th}$ term is called the general term:

General Term Formula

$$T_{r+1} = \binom{n}{r} a^{n-r} b^r$$

where r = 0, 1, 2, ..., n

Example: Find 3rd term in $(x - 2y)^5$

Step 1: For 3rd term, r + 1 = 3 ⇒ r = 2

Step 2: a = x, b = -2y, n = 5

Step 3: Apply formula:

$T_3 = \binom{5}{2} x^{5-2} (-2y)^2$

Step 4: Calculate:

$\binom{5}{2} = 10$, $(-2y)^2 = 4y^2$

Step 5: $T_3 = 10 \cdot x^3 \cdot 4y^2 = 40x^3y^2$

5. Practice Problems

📝 Basic Practice

1. Expand $(x + 2)^4$ using binomial theorem

Hint: Use a = x, b = 2, n = 4

2. Find the 4th term in the expansion of $(2a - b)^6$

Hint: Remember T₄ means r+1=4 ⇒ r=3

3. Write the expansion of $(1 + x)^5$ using Pascal's triangle

Hint: Row n=5: 1, 5, 10, 10, 5, 1

🎯 JEE Level Practice

4. Find the coefficient of $x^3$ in $(2 + 3x)^8$

Hint: Use general term and equate power of x to 3

5. If the 3rd term in $(1 + x)^n$ is 36x², find n

Hint: T₃ = ⁿC₂ x² = 36x²

📋 Quick Summary

Key Formulas

$(a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$
General term: $T_{r+1} = \binom{n}{r} a^{n-r} b^r$
Binomial coefficient: $\binom{n}{r} = \frac{n!}{r!(n-r)!}$
Number of terms: $n + 1$

Properties to Remember

Sum of coefficients: Put a = b = 1
Middle term: Depends on whether n is even or odd
Greatest coefficient: For (1 + x)ⁿ, middle term(s) have max coefficient
Pascal's identity: $\binom{n}{r} + \binom{n}{r-1} = \binom{n+1}{r}$

Ready for Advanced Binomial Theorem?

Learn about middle terms, greatest coefficients, and binomial theorem for rational indices

Advanced Binomial Theorem → More Algebra Topics