Number and Algebra - Binomial theorem (introduction)

A binomial is an algebraic expression containing two terms, such as $(a + b)$. The Binomial Theorem provides a method to expand powers of these expressions, like $(a + b)^n$, without performing repeated long-form multiplication.

Pascal's Triangle is a visual geometric arrangement where each number is the sum of the two numbers directly above it. It starts with a 1 at the top (Row 0). Row 1 is 1, 1; Row 2 is 1, 2, 1; and Row 3 is 1, 3, 3, 1. The numbers in the $n^{th}$ row correspond exactly to the coefficients in the expansion of $(a + b)^n$.

Factorial notation, written as $n!$, represents the product of all positive integers from 1 up to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. It is defined that $0! = 1$. This calculation is a building block for finding binomial coefficients.

Binomial coefficients, denoted as $\binom{n}{r}$ or $^nC_r$, represent the values found in Pascal's Triangle. The notation $\binom{n}{r}$ (read as 'n choose r') identifies the coefficient for the term containing $b^r$ in the expansion of $(a + b)^n$.

In any binomial expansion $(a + b)^n$, the powers of the first term $a$ decrease from $n$ to 0 across the terms, while the powers of the second term $b$ increase from 0 to $n$. Visually, you can track this as $a^n, a^{n-1}b, a^{n-2}b^2, \dots, b^n$.

The sum of the exponents of $a$ and $b$ in every single term of the expansion of $(a + b)^n$ must always equal the original power $n$. For example, in $(x + y)^5$, the term containing $x^2$ must also contain $y^3$ because $2 + 3 = 5$.

The total number of terms in the expansion of $(a + b)^n$ is always $n + 1$. For instance, $(a + b)^2$ expands to $a^2 + 2ab + b^2$, which contains $2 + 1 = 3$ terms.