Binomial Expansion Pyramid Quiz and Flashcards

Study Notes

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

Description

Learn about Pascal's Triangle, its construction, properties, and applications in binomial expansion and combinations. Discover the symmetry and row sums of the triangle, and how it's used to calculate binomial coefficients.