Combinations and Permuations – Counting Methods

I wrote a couple of posts about combinations and permutations. These posts included definitions and examples of working with both permutations and combinations, but I know reading them is not always the best way for students to learn, so here is a video lesson of the two counting methods: combinations and permutations.



Any mathematics textbook you look you will find a definition of a permutation.The thing is, no matter where you look, they will all be the same. Permutations and Combinations are a ways to count a sample space.  There is a major difference between the two.

Permutations are derived from the fundamental counting principal, but you will choose all of the items in the set. For example if you have 5 books to put on a shelf. For the first book to shelf you 5 to choose from and you shelf one of them. Now you have 4 to choose from and you shelf one of them. Now you have 3 to choose from and you shelf one of them. Now there are 2 and you shelf one of them and now you only have 1 to shelf. Well, the fundamental counting principal would give:

Example: 5! or 5 Factorial

Example: 5! or 5 Factorial

This type of multiplication is referred to as a factorial. The above example would be read as “five factorial” and written as 5!. But life is full of choices and many teams have more players than starters. Knowing how many ways you can arrange a team can be helpful with coaching decisions. Do you know how many ways a coach can arrange 5 starters on a basketball team when there are 11 players to choose from? It is not a guessing game either. If you don’t know that is okay. Think about the only two numbers mentioned in the problem. 11 players to choose from but only 5 starting spots. Below is the formal definition of the number of permutations.

The number of permutations of n items of a set arranged r items at a time is

Definition of Permuations

Definition of Permutations

The great thing about living in today’s information age is the technology. It is every where. The nice thing about technology is the pain staking task of finding permutations can now be done by calculator. Solving these types of problems could not be easier. Suppose you there were 14 wrestlers competing in a tournament and the top 4 places were awarded a trophy and prize money. In this situation, the order in which the wrestlers is important to all parties involve. The organizers need to know who to pay and the wrestlers care where they place because the higher the place the higher the prize money. You would count the different ways the 14 wrestlers could finish in the top 4 using a permutation. P(14, 4) = 24, 024 ways the wrestlers could finish.

There are many other situations you could create or may even run into, where counting the sample space is necessary or fun! Soon to come, a post about combinations and a video on both permutations and combinations.

%d bloggers like this: