Binomial Expansion Using Pascal's Triangle

A binomial is an expression of the form a + b, where a and b are numbers or algebraic expressions.

Raising it to a power n produces a series of terms that follow a predictable pattern.

Each expansion has n + 1 terms, starting with aⁿ and ending with bⁿ.

These patterns align with a visual tool known as Pascal’s Triangle.

Pascal’s Triangle is a triangular array where each row provides the coefficients for a binomial raised to a specific power.

For example, the fifth row gives the coefficients for a + b raised to the fifth power.

Each row begins and ends with one, and each inner number equals the sum of the two numbers diagonally above it. Pascal’s triangle provides the coefficients of the Binomial Theorem, calculated as n choose k.

This pattern also applies to probability. In coin tosses, H and T represent heads and tails. For three tosses, the sum of H and T raised to the third power represents all possible outcomes.

After expanding and comparing with Pascal’s triangle, each term corresponds to a possible outcome: three heads, two heads and one tail, one head and two tails, or three tails.