### 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