Combination and Permutation

Overview

Combination and permutation are two different way to arrange a set of items. The distinction between the two is

• Combination: Order does not matter (e.g., $ABC=BAC$)
• Permutation: Order matters (e.g., $ABC\ne BAC)$

Combination

Combinations refer to the different ways to choose a subset of items from a larger set, where the order does not matter. The formula for combination is given by

$$C(n,r)=\left(\genfrac{}{}{0ex}{}{n}{r}\right)=\frac{n!}{r!(n-r)!}$$

where

• $n$: Total number of items
• $r$: Number of items to arrange

Example: Forming a committee from a group

You have a group of 5 people, $A,B,C,D,E$, in which you want to form a committee of 3 people. How many committees of 3 people can you create from these 5 people?

In this case, there are a total of 5 items $(n=5)$, and 3 items to arrange $(r=3)$. So, the combination can be obtained by:

$$C(n,r)=C(5,3)=\frac{5!}{3!(5-3)!}=\frac{120}{6\cdot 2}=10$$

So, there are 10 different ways to form a committee from a group of 5 peole

Permutation

Permutations refer to the different ways to arrange a set of items, where the order matters. The formula for permutation is given by

$$P(n,r)=\frac{n!}{(n-r)!}$$

Example: Ordering books

You want to arrange 2 books out of $A,B,C$ on a shelf. How many different ways can you do it? Note in permutation, that order of the books matters (e.g., “$A,B,C\ne B,A,C$). In this case, $n=3$, and $r=2$. So, the number of permutations is given by