Binomial Theorem: Pascal's Triangle

Binomial Theorem: Pascal's Triangle

Definition

Pascal's Triangle is a triangular array of binomial coefficients, where each number is the number of combinations of a certain size that can be selected from a set of items.

Construction

• The triangle is constructed by starting with a 1 at the top, and each subsequent row is generated by adding the two numbers directly above it to get the number below.
• Each row represents the binomial coefficients of a binomial expansion, where the top row represents the zeroth power, and each subsequent row represents a higher power.

Properties

• Symmetry: The triangle is symmetric, meaning the numbers on each row are the same when read forwards and backwards.
• Binomial Coefficients: Each number in the triangle is a binomial coefficient, representing the number of combinations of a certain size that can be selected from a set of items.
• Row Sums: The sum of the numbers in each row is a power of 2 (2^n, where n is the row number).

Applications

• Binomial Expansion: Pascal's Triangle is used to expand binomials of the form (a+b)^n, where n is a positive integer.
• Combinations: The triangle is used to calculate the number of combinations of a certain size that can be selected from a set of items.

Example

• The 4th row of Pascal's Triangle is: 1 4 6 4 1
• These numbers represent the binomial coefficients for (a+b)^3: (a+b)^3 = a^3 + 4a^2b + 6ab^2 + 4b^3 + b^3