# Binomial Coefficient

$$\binom{n}{k}=\dfrac{n!}{k!(n-k)!}$$
The binomial coefficient is sometimes written $^nC_k$ or $C^n_k$, depending on authors or geographical regions.

\begin{aligned} \binom{n}{k} &= \dfrac{n!}{k!(n-k)!} \cdots (1) \\ \binom{n}{n-k} &= \dfrac{n!}{(n-k)!(n-(n-k))!} = \dfrac{n!}{(n-k)!k!} \cdots (2) \\ \therefore \binom{n}{k} &= \binom{n}{n-k} \text{by } (1) \text{ and } (2) \end{aligned}

This means;
\begin{aligned} \binom{10}{2} &= \binom{10}{8} \\ \binom{100}{1} &= \binom{100}{99} \end{aligned}

The following binomial coefficients are undefined.

$\displaystyle \binom{5}{7} = \dfrac{5!}{7! \times (5-7)!} \rightarrow (-2)!$ is undefined

$\displaystyle \binom{5}{3.3} = \dfrac{5!}{3.3! \times (5-3.3)!} \rightarrow 3.3!$ is undefined

$\displaystyle \binom{5.2}{3} = \dfrac{5.2!}{5.2! \times (5.2-3)!} \rightarrow 5.2!$ is undefined

$\displaystyle\binom{5}{-2} = \dfrac{5!}{(-2)! \times (5-(-2))!} \rightarrow (-2)!$ is undefined

## Relationship with Pascal’s triangle

$\begin{matrix} n=0&&&&&&1 \\ &&&&&&\displaystyle\binom{0}{0} \\ n=1&&&&&1&&1 \\ &&&&&\displaystyle \binom{1}{0}&& \displaystyle\binom{1}{1} \\ n=2&&&&1&&2&&1 \\ &&&&\displaystyle \binom{2}{0} &&\displaystyle \binom{2}{1} &&\displaystyle \binom{2}{2} \\ n=3&&&1&&3&&3&&1 \\ &&&\displaystyle \binom{3}{0} &&\displaystyle \binom{3}{1} &&\displaystyle \binom{3}{2} &&\displaystyle \binom{3}{3} \\ n=4&&1&&4&&6&&4&&1 \\ &&\displaystyle \binom{4}{0} &&\displaystyle \binom{4}{1} &&\displaystyle \binom{4}{2} &&\displaystyle \binom{4}{3} &&\displaystyle \binom{4}{4} \end{matrix}$

## Example 1

Simplify $\displaystyle \binom{n}{n-1}$.

\begin{align} \displaystyle \binom{n}{n-1} &= \dfrac{n!}{(n-1)! \times 1!} \\ &= \dfrac{n \times (n-1)!}{(n-1)!} \\ &= n \end{align}

## Example 2

Evaluate $\displaystyle \binom{8}{2}$.

\begin{align} \displaystyle \binom{8}{2} &= \dfrac{8!}{2! \times 6!} \\ &= \dfrac{8 \times 7 \times 6!}{2 \times 1 \times 6!} \\ &= \dfrac{8 \times 7}{2} \\ &= 28 \end{align}

## Example 3

Evaluate $\displaystyle \binom{8}{0}$.

\begin{align} \displaystyle \binom{8}{0} &= \dfrac{8!}{0! \times 8!} \\ &= \dfrac{8!}{1 \times 8!} \\ &= 1 \end{align}

## Example 4

Evaluate $\displaystyle \binom{4}{5}$.

\begin{align} \displaystyle \binom{4}{5} &= \dfrac{4!}{5! \times (4-5)!} \\ &= \dfrac{4!}{5! \times (-1)!} \\ &= \text{undefined} \end{align}

âœ“ Discover more enlightening videos by visiting our YouTube channel!

## Mastering Integration by Parts: The Ultimate Guide

Welcome to the ultimate guide on mastering integration by parts. If you’re a student of calculus, you’ve likely encountered integration problems that seem insurmountable. That’s…

## Binomial Expansions

The sum $a+b$ is called a binomial as it contains two terms.Any expression of the form $(a+b)^n$ is called a power of a binomial. All…

## The Best Practices for Using Two-Way Tables in Probability

Welcome to a comprehensive guide on mastering probability through the lens of two-way tables. If you’ve ever found probability challenging, fear not. We’ll break it…

## High School Math for Life: Making Sense of Earnings

Salary Salary refers to the fixed amount of money that an employer pays an employee at regular intervals, typically on a monthly or biweekly basis,…