Number and Algebra - The Binomial Theorem

The Binomial Theorem provides a method for expanding expressions of the form $(a + b)^n$, where $n$ is a positive integer. The expansion consists of a sum of $n + 1$ terms where the powers of $a$ decrease from $n$ to $0$ while the powers of $b$ increase from $0$ to $n$.

Pascal's Triangle is a visual geometric arrangement of binomial coefficients. Imagine a triangle starting with a $1$ at the top (Row $0$). Each subsequent row starts and ends with $1$, and every interior number is the sum of the two numbers directly above it. The $n$-th row of this triangle provides the coefficients $\binom{n}{r}$ for the expansion of $(a + b)^n$.

Binomial Coefficients, denoted as $\binom{n}{r}$ or $nCr$, represent the number of ways to choose $r$ items from a set of $n$ items. Visually, these coefficients are symmetrical across the center of each row in Pascal's Triangle, meaning $\binom{n}{r} = \binom{n}{n-r}$.

The General Term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ is used to find a specific term in an expansion without writing out the entire series. Note that the $(r+1)$-th term uses the value $r$ in the combination and the exponent of the second variable.

Factorial notation ($n!$) is the product of all positive integers up to $n$. It is the mathematical engine used to calculate binomial coefficients. Visually, factorials grow extremely rapidly as $n$ increases, which is why binomial coefficients in the middle of the expansion (like the peak of a bell curve) are much larger than those at the ends.

In every term of the expansion $(a+b)^n$, the sum of the exponents of $a$ and $b$ must always equal $n$. For example, in the expansion of $(a+b)^5$, a term might be $k a^2 b^3$, where $2 + 3 = 5$.

The sum of all binomial coefficients in the $n$-th row of Pascal's Triangle is equal to $2^n$. This can be visualized by summing the horizontal rows: Row 0 is $1 (2^0)$, Row 1 is $1+1=2 (2^1)$, Row 2 is $1+2+1=4 (2^2)$, and so on.