Notes from Toppers

Binomial Expansions

1. Binomial Theorem

  • Formula: (a + b)^n = ∑C(n, r) * a^(n-r) * b^r, where C(n, r) is the binomial coefficient.
  • Binomial Coefficients: C(n, r) = n!/(r!(n-r)!)
  • Properties:
    • Symmetry: C(n, r) = C(n, n-r)
    • Pascal’s Triangle: C(n, r) can be represented in the form of Pascal’s triangle.

2. Applications in Combinations and Probability

  • Combinations: Binomial expansion is used to count the number of ways to select r objects from a set of n distinct objects.
  • Probability: Binomial expansion is used to calculate probabilities in various scenarios, such as the binomial distribution and the normal distribution.

3. Power Series Expansions

  • Concept: A power series expansion is an infinite series of terms involving powers of a variable x.
  • Examples:
    • sin(x) = x - x^3/3! + x^5/5! - …
    • cos(x) = 1 - x^2/2! + x^4/4! - …
    • e^x = 1 + x + x^2/2! + x^3/3! + …
    • ln(1+x) = x - x^2/2 + x^3/3 - …

4. Binomial Approximations

  • Approximation: (1+x)^n ≈ 1 + nx when x is small compared to 1.
  • Applications:
    • Approximating probabilities in the binomial distribution
    • Simplifying complex expressions

5. Multinomial Expansions and Generalizations

  • Multinomial Theorem: (a + b + c)^n = ∑C(n, r, s) * a^r * b^s * c^t, where C(n, r, s, t) is the multinomial coefficient.
  • Multinomial Distribution: The multinomial distribution is a generalization of the binomial distribution for multiple categories.

6. Series involving Binomial Expansions

  • Sum of Finite Terms: ∑C(n, r) * a^r * b^(n-r) = (a + b)^n
  • Infinite Series: ∑C(n, r) * x^r diverges for |x|>1 and converges for |x|<1.

7. Applications in Calculus

  • Derivatives: Binomial expansion can be used to find derivatives of certain functions.
  • Integrals: Binomial expansion can be used to find integrals of certain functions.

References:

  • NCERT Mathematics, Class 11, Chapter 15:Binomial Theorem
  • NCERT Mathematics, Class 12, Chapter 9:Sequences and Series


Table of Contents