📈 Binomial Theorem

1. Introduction

The Binomial Theorem provides a formula for expanding a binomial expression raised to any positive integral power without doing the manual multiplication.

2. Statement of the Binomial Theorem

For any positive integer n:

(x + y)ⁿ = ⁿC₀ xⁿ + ⁿC₁ xⁿ⁻¹y + ⁿC₂ xⁿ⁻²y² + ... + ⁿCₙ yⁿ Where ⁿC_r are binomial coefficients.

Key Observations

1. The total number of terms in the expansion is (n + 1).

2. The sum of the indices of x and y in every term is exactly n.

3. Binomial coefficients equidistant from beginning and end are equal (ⁿCr = ⁿC(n-r)).

3. General and Middle Terms

General Term: The (r+1)th term is denoted by T_(r+1).

T_(r+1) = ⁿCr xⁿ⁻ʳ yʳ

Finding the Middle Term

If n is even: There is only one middle term, which is the (n/2 + 1)th term.

If n is odd: There are two middle terms, the ((n+1)/2)th term and the ((n+3)/2)th term.

4. Special Cases

If x = 1 and y = x: (1 + x)ⁿ = ⁿC₀ + ⁿC₁x + ⁿC₂x² + ... + ⁿCₙxⁿ

Sum of all binomial coefficients (put x=1, y=1): ⁿC₀ + ⁿC₁ + ... + ⁿCₙ = 2ⁿ.