Binomial Theorem

Board: CBSE | Class: Class 11

Comprehensive study notes for Binomial Theorem by Ajay Yadav (Math King of Katargam). Master every concept with clear explanations, solved examples, and practice problems.

Key Concepts

Binomial Theorem

(a+b)ⁿ = ⁿC₀aⁿ + ⁿC₁aⁿ⁻¹b + … + ⁿCₔaⁿ⁻ₔbₔ + … + ⁿCₙbⁿ. Total n+1 terms.

General Term

Tₔ₍₂ = ⁿCₔ aⁿ⁻ₔ bₔ, where r = 0, 1, 2, …, n. This is the (r+1)th term.

Middle Term

If n is even: middle term = Tₔ/₂₍₂ = ⁿCₙ/₂ aⁿ/₂ bⁿ/₂. If n is odd: two middle terms = Tₔ₍₂/₂ and Tₔ₍₂₍₂/₂.

Binomial Coefficients

Coefficients ⁿC₀, ⁿC₁, …, ⁿCₙ have properties: Sum = 2ⁿ. Sum of even-positioned = Sum of odd-positioned = 2ⁿ⁻¹.

Pascal's Triangle

Each row: 1. Each subsequent entry = sum of two above. Row n gives coefficients of (a+b)ⁿ. Row 0: 1. Row 1: 1 1. Row 2: 1 2 1. Row 3: 1 3 3 1.

Important Formulas

General Term	Tₔ(2 = ⁿCₔaⁿ⁻ₔbₔ
Middle (n even)	Tₔ/2(2 = ⁿCₙ/2aⁿ/2bⁿ/2
Sum of Coefficients	ⁿC₀ + ⁿC1 + ... + ⁿCₙ = 2ⁿ
Specific Term	Find r such that power of x matches desired exponent

Solved Examples

Example 1: Expand (x+2)⁴ using binomial theorem.

Solution: (x+2)⁴ = ⁴C₀x⁴ + ⁴C₁x³(2) + ⁴C₂x²(4) + ⁴C₁x(8) + ⁴C₀(16) = x⁴ + 8x³ + 24x² + 32x + 16.

Example 2: Find the 5th term in (2x-3)⁶.

Solution: T₅ = Tₔ₍₂ with r=4. T₅ = ⁶C₄(2x)⁶⁻₄(-3)⁴ = 210(2x)⁰(-3)⁴ = 210(1)(81) = 17010.

Example 3: Find the middle term in (x + 1/x)⁶.

Solution: n=10 even. Middle = T₆ (r=5). T₆ = ⁶C₅x⁶⁻₅(1/x)₅ = 252(x⁵)(1/x⁵) = 252.

Practice Questions

1. Expand (x – 1/x)⁴.
2. Find the coefficient of x³ in (2x+3)⁵.
3. Find the term independent of x in (x² + 1/x³)⁶.
4. Using binomial theorem, find (0.99)⁴ correct to 4 decimal places.
5. Prove that ⁿC₀ + ⁿC₂ + ⁿC₄ + … = 2ⁿ⁻¹.

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