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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 15,600 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 212,520 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 3,628,800 different orders.