# Binomial Theorem

*See also:*binomial coefficient, factorial

The *Binomial Theorem* is an essential algebraic manipulation tool widely used in practical applications. The theorem states that for any given positive integer \(n\), the expansion of the binomial expression \((a+b)^n\) can be expressed as the sum of \(n+1\) terms, where each term is a coefficient multiplied by the product of the two binomial expressions \(a\) and \(b\), each raised to a power.

This theorem has numerous applications in fields such as probability theory, statistics, and calculus and is an essential concept for any student or practitioner of mathematics to understand. In short, the formula is represented by the following expression:

\[(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \ldots + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n \]

The formula is composed of the following elements:

- \(n\), the exponent term, is a positive integer \(n \in \mathbb{Z}\).
- \(a\) is raised to a decreasing power, from \(n\) to \(0\).
- \(b\) is raised to an increasing power from \(0\) to \(n\)
- The term \(\binom{n}{k}\) is the binomial coefficient, where \(k\) takes values between \(0\) and \(n\).

## Example

The cube of a binomial is an example of the application of the binomial theorem. Consider the cube of a binomial \((a + b)^3\). Applying the formula to the cube of a binomial \((a + b)^3\), substituting \(n=3\) we get:

\[(a + b)^3 = \binom{3}{0} a^3 b^0 + \binom{3}{1} a^2 b^1 + \binom{3}{2} a^1 b^2 + \binom{3}{3} a^0 b^3 \]

The binomial coefficient \(\binom{n}{k}\) represents the number of ways to select \(k\) elements from a set of \(n\) elements without considering their order. The formula is summarized briefly below:

\[\binom{n}{k} = \frac{n!}{k!(n-k)!} \]

- \(n!\) and \(k!\), \(n\) and \(k!\) factorial, represent the product of all positive integers from \(1\) to \(n\) and from \(1\) to \(k\).
- \((n-k)!\), \(n-k\) factorial represents the product of all positive integers from \(1\) to \(n-k\).

For example:

\[\begin{align*} \binom{3}{2} &= \frac{3!}{2!(3-2)!} \\[0.8em] &= \frac{3 \cdot 2 \cdot 1}{2 \cdot 1 \cdot (3-2)} \\[0.8em] &= \frac{6}{2} \\[0.8em] &= 3 \end{align*}\]

Simplifying the expression we obtain:

\[(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\]

So, the cube of a binomial \((a + b)^3\), expanded using the binomial theorem, gives us the notable product \(a^3 + 3a^2b + 3ab^2 + b^3\).