Binomial Theorem

 

Binomial Theorem: A Quick Overview

The Binomial Theorem provides a formula for expanding a binomial expression raised to a positive integer power. It's a fundamental tool in algebra and combinatorics.

The Formula:

For any positive integer 'n' and any real numbers 'a' and 'b':

(a + b)^n = ∑(nCr) * a^(n-r) * b^r

where:

• ∑ is the summation symbol
• nCr is the binomial coefficient, also known as 'n choose r'
• r ranges from 0 to n

Understanding the Terms:

• Binomial coefficient (nCr): This represents the number of ways to choose 'r' objects from a set of 'n' objects. It can be calculated using the formula:

  nCr = n! / (r! * (n-r)!)
  

  where n! is the factorial of n.
• a^(n-r) and b^r: These are the powers of 'a' and 'b' in each term of the expansion.

Example:

Let's expand (x + 2)^3 using the Binomial Theorem:

(x + 2)^3 = 3C0 * x^3 * 2^0 + 3C1 * x^2 * 2^1 + 3C2 * x^1 * 2^2 + 3C3 * x^0 * 2^3

Simplifying this expression, we get:

(x + 2)^3 = x^3 + 6x^2 + 12x + 8

Key Points:

• The coefficients in the expansion are given by the binomial coefficients.
• The powers of 'a' and 'b' in each term add up to 'n'.
• The expansion has 'n+1' terms.

Applications of the Binomial Theorem:

• Finding powers of binomials
• Deriving formulas for various mathematical expressions
• Probability theory
• Combinatorics