Binomial Theorem
Board: JEE Main | Exam: JEE Main
JEE Main study notes for Binomial Theorem by Ajay Yadav. This chapter covers all concepts required for JEE Main with problem-solving techniques.
Key Concepts
General Expansion
(a+b)ⁿ = Σₔ≈₀ⁿ ⁿCₔaⁿ⁻ₔbₔ. Total n+1 terms.
General Term
Tₔ(2 = ⁿCₔaⁿ⁻ₔbₔ. Middle terms. Numerically greatest term.
Binomial Coefficients
ⁿC₀+ⁿC1+…+ⁿCₙ=2ⁿ. ⁿC₀+ⁿC2+ⁿC₄+…=2ⁿ⁻¹. ⁿC1+ⁿC₃+ⁿC₅+…=2ⁿ⁻¹.
Multinomial Theorem
(x1+x2+…+xₔ)ⁿ = Σ (n!/(n₁!n₂!…nₔ!)) x1ⁿ¹x2ⁿ²…xₔⁿₔ.
Important Formulas
| General term | Tₔ(2 = ⁿCₔaⁿ⁻ₔbₔ |
| Coefficient sum | ΣⁿCₔ = 2ⁿ |
| Odd/even sum | Each = 2ⁿ⁻¹ |
Solved Examples
Example 1: Coefficient of x³ in (x+2)⁴.
Solution: Tₔ(2 with r=3: ⁴C1x(8) = 32. 32.
Practice Questions
- Find term independent of x in (x²+1/x)⁶.
- Find (1.02)⁴ using binomial.
- Find coefficient of x⁶ in (1+x)⁸+(1+x)⁶.
- Find sum ⁿC1+2ⁿC2+3ⁿC₃+…+nⁿCₙ.
- Find the numerically greatest term in (3+2x)⁶ when x=1.