Understanding the Binomial Theorem

Oct 30, 2024

Binomial Theorem Lecture Notes

Overview

  • Topic: The Binomial Theorem
  • Structure: 3 parts
    1. Binomial Coefficient Symbol
    2. Pascal's Triangle
    3. Binomial Theorem

1. Binomial Coefficient Symbol

Definition

  • Symbol: ( \binom{n}{j} ) (read as "n choose j" or "n taken j at a time")
  • Formula: ( \binom{n}{j} = \frac{n!}{j!(n-j)!} )
  • Restrictions: 0 ≤ j ≤ n; n is a positive integer

Important Notes

  • Common notation: Sometimes written as ( nCj ) (C stands for combinations)
  • On calculators: Look for a key labeled NCR or similar
  • Common Misconception: ( \binom{n}{j} ) is NOT ( \frac{n}{j} )

Examples

  1. Example: Calculate ( 3 \text{ choose } 1 )
    • ( \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{3}{1} = 3 )
  2. Example: Calculate ( 65 \text{ choose } 15 )
    • ( \binom{65}{15} = \frac{65!}{15!(65-15)!} = \frac{65!}{15!50!} )
    • Simplification by factoring out factorials.

Properties

  • Symmetry: ( \binom{n}{j} = \binom{n}{n-j} )
  • Special Cases:
    • ( \binom{n}{0} = 1 )
    • ( \binom{n}{n} = 1 )
    • ( \binom{n}{1} = n )
    • ( \binom{n}{n-1} = n )

2. Pascal's Triangle

Structure

  • Created using the binomial coefficients.
  • Example of first few rows:
    • Row 0: 1
    • Row 1: 1, 1
    • Row 2: 1, 2, 1
    • Row 3: 1, 3, 3, 1
    • Row 4: 1, 4, 6, 4, 1

Properties

  • Vertical Symmetry: Each row is symmetric.
  • Each row starts and ends with 1.
  • Row Generation Rule: Each element is the sum of the two elements directly above it.

Historical Context

  • Known to mathematicians before Pascal, including:
    • Jordanus de Namur (1225 AD)
    • Yang Hui (1261 AD)
    • Chu (1303 AD)

3. Binomial Theorem

Theorem Statement

  • Expansion: ( (x + a)^n = \sum_{j=0}^{n} \binom{n}{j} x^{n-j} a^j )

Key Observations

  • Powers of x and a add up to n.
  • Coefficients correspond to binomial coefficients in Pascal's triangle.

Example Expansions

  1. Example: Expand ( (x + 2)^5 )

    • Using the binomial theorem: ( (x + 2)^5 = \sum_{j=0}^{5} \binom{5}{j} x^{5-j} (2)^j )
    • Calculate coefficients and powers:
      • ( 5 \text{ choose } j )

  2. Example: Expand ( (2y - 3)^4 )

    • Rewrite as ( (2y + (-3))^4 ) and use the binomial theorem.

Finding Specific Terms

  • To find the coefficient of a specific power of a variable:
    • Identify which term corresponds to that power using the general term formula.

Challenge Problem

  • Compare ( 1000^{1000} ) and ( 1001^{999} ) using the binomial theorem.

Summary

  • The Binomial Theorem allows for efficient expansion of binomials raised to powers.
  • The connection between the binomial coefficients and Pascal's Triangle is key to understanding the theorem.

Full transcript