Combinations and Permutations

PERMUTATIONS AND COMBINATIONS

After mentioning permutations and combinations in a previous blog post on targeting, I thought it was time to provide a more mathematical treatment of them.  Everybody talks about them as a pair (in the same way as people tend to say 'look and feel', or 'design and technology').  

Let's start with an example:  three banners are to be shown on a website homepage. If we simplify and call the different pictures A, B and C, then one order in which they can be hung is A, B, C and another is A, C, B.

Each of these arrangements is called a permutation of the three pictures (and there are further possible permutations), i.e, a permutation is an ordered arrangement of a number of items.

Suppose, however, that seven banners are available for presenting on the website, and only three of them can be displayed. This time a choice has first to be made. If we call the seven banners A, B, C, D, E, F and G, one possible choice of the three pictures for display is A, B, and C - ignoring the sequence of the banners. Regardless of the order in which they are then hung this group of three is just one choice and is called a combination.

A, B, C
A, C, B
B, A, C
B, C, A
C, A, B
C, B, A

are six different permutations; but only one combination thus:

i.e. a combination is an unordered selection of a number of items from a given set.

In this post,  I will discuss methods for finding the total number of ways of arranging items (permutations) or choosing groups of items (combinations) from a given set. But before we do so it is critical that we're able to distinguish between permutations and combinations.  They are not the same, and the terms shouldn't be used interchangeably.

For example:  a news website has ten news articles on its site, but the home page layout means that only five can be shown, in a vertical column. While they cannot display all ten of the articles, they must choose a group of five. The order in which the site selects the five articles is irrelevant (in this case); the set of five is only one combination. Once they have made the choice, they are then able to place the five articles in various different orders on the display stand. Now the site team are arranging them and each arrangement is a permutation, i.e a particular set of five articles is one combination, but that one combination can be arranged to give several different permutations.

1.  The King's Health is Failing
2.  Peace Treaty Signed!
3.  Life found on Mars!
4. Bungled Theft on the Railway
5. Jack the Ripper
6. Reports of My Death Greatly Exaggerated
7. Lottery Winner Buys Football Team
8.  New 007 is a Woman
9. Crop Circles - The Answer
10. Price of Eggs falls 10%

In each of these examples, decide if the question is asking for a number of permutations, or a number of combinations.

How many arrangements of the letters A, B, C are there?
Arrangements means the sequence is important, so this means permutations.

A team of six members is chosen from a group of eight. How many different

teams can be selected?
The sequence is not important, so this means combinations.

A person can take eight records to a desert island, chosen from his own

selection of one hundred records. How many different sets of records could he choose?
Different sets, again the sequence is not critical, so these are combinations.

The first, second and third prizes for a raffle are awarded by drawing tickets

from a box of five hundred. In how many ways can the prizes be won?
Here, there's a difference between the order (or sequence, or arrangement) of the three prizes, so we're looking at permutations.

Combinations:  the sequence is not important.
Permutations:  the sequence is important.