# Permutation Examples

## Example 1

How many three letter sequences are there, assuming that letters cannot be
repeated? The order in which the letters appear is significant (e.g.
"bad" and "dab" are different sequences). This problem can
be stated as, "Find the number of permutations of 26 things taken 3 at a
time."

**1. Determine the number of decisions.**

There are three decisions since each letter in the three-letter sequence
corresponds to a decision.

**2. For each decision, determine the number of options.**

For the first letter there are 26 options. For the second there are 25
options since no repeats are allowed. For the third letter there are 24 options.

**3. Multiply these numbers together to determine the number of
arrangements.**

There are
three-letter sequences.

## Example 2

The Asbury Fun & Games Club has 23 members. The club plans to hold an
election for a president, a vice president, a secretary, and a treasurer. In how
many different ways can these four offices be filled? The order is important
because of the context of the problem. Each person is running for a specific
office and switching offices results in a different slate (even if it is the
same two people). Find the permutations of 23 things taken 4 at a time.

**1. Determine the number of decisions.**

There are four decisions; one for each office.

**2. For each decision, determine the number of options.**

Any one of the 23 club members could hold the office of president. That
leaves 22 members who could fill the office of vice president. There are 21
members left who could be chosen as secretary. Any one of the remaining 20 could
be selected treasurer.

**3. Multiply these numbers together to determine the number of
arrangements.**

The four offices could be filled in
different ways.

## Example 3

Ten people are being honored at an awards banquet. In how many different
orders can the awards be presented? The problem is to find the
permutations of 10 things taken 10 at a time.

**1. Determine the number of decisions.**

There are ten decisions; who goes first, who goes second, and so on.

**2. For each decision, determine the number of options.**

Any one of the 10 honorees could be selected to go first. That leaves 9 to go
second, 8 to go third, and so on. By the time the decision is made on who goes
last, there is only one honoree remaining.

**3. Multiply these numbers together to determine the number of
arrangements.**

The awards could be given in
different orders.