These are also called arrangements because the order in which the elements are selected matters.
Permute $r$ out of $n$ objects
Given $n$ objects, the number of arrangements of $r$ objects is: $$ \begin{equation}\begin{aligned} {}^n P_r =\frac{n!}{(n-r)!}\\ \end{aligned}\end{equation} $$
Example: Your mom bought $6$ toys for you but you only have $3$ boxes. How many ways can you arrange the toys into the $3$ boxes?
Solution: $$ \begin{equation}\begin{aligned} {}^6 P_3&=\frac{6!}{(6-3)!}\\ &=\frac{6!}{3!}\\ &=120\\ \end{aligned}\end{equation} $$
Explanation:
- There are $ 6$ possible selections for item $1$
- There are $5$ possible selections for item $2$
- There are $4 $ possible selections for item $3$
Permute $n$ out of $n$ objects
Given $n$ objects, the number of arrangements of all $n$ objects is: $$ \begin{equation}\begin{aligned} {}^n P_n &=\frac{n!}{(n-n)!}\\ &=\frac{n!}{0!}\\ &=\frac{n!}{1}\\ &=n!\\ \end{aligned}\end{equation} $$
Example: Determine how many arrangements we can have for $5$ paintings.
Solution: $$ \begin{equation}\begin{aligned} {}^5 P_5 &=5!\\ &=120\\ \end{aligned}\end{equation} $$
Explanation:
- There are $ 5$ possible selections for item $1$
- There are $4$ possible selections for item $2$
- There are $3$ possible selections for item $3$
- There are $2$ possible selections for item $4$
- There are $1 $ possible selections for item $5$
Permutations with repetitions
For $n$ elements and $p$, $q$, etc. repetitions, the number of permutations is given as: $$ \begin{equation}\begin{aligned} =\frac{n!}{p!\times q!\times ...}\\ \end{aligned}\end{equation} $$
Example: Find the number of arrangements for the letters in the word MISSISSIPPI.
Solution: There are $11$ letters in the word. There are $4$ S’s, $4$ I’s and $2$ P’s. Thus the number of permutations is: $$ \begin{equation}\begin{aligned} &=\frac{11!}{4!\times 4!\times 2!}\\ &=34,650\\ \end{aligned}\end{equation} $$