Binomial Coefficient

combinatorics

Definition

Binomial Coefficient

The Binomial Coefficient, denoted as $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ or “n choose k,” is not a probability distribution, but rather a mathematical expression that counts the number of ways to choose $k$ elements out of a total of $n$ distinct elements without regard to order.

$$\underset{\text{n choose k}}{\underbrace{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}}=\frac{n!}{k!(n-k)!}$$

Properties

$$\left(\genfrac{}{}{0ex}{}{n}{0}\right)=\left(\genfrac{}{}{0ex}{}{n}{n}\right)=1$$ $$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\left(\genfrac{}{}{0ex}{}{n}{n-k}\right)$$ $$\left(\genfrac{}{}{0ex}{}{n+1}{k+1}\right)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)+\left(\genfrac{}{}{0ex}{}{n}{k+1}\right)$$