A Permutation is an ordered Combination.
 | To help you to remember, think "Permutation ... Position" |
Permutations
There are basically two types of permutation:
- Repetition is Allowed: such as the lock above. It could be "333".
- No Repetition: for example the first three people in a running race. You can't be first and second.
1. Permutations with Repetition
These are the easiest to calculate.
When you have n things to choose from ... you have n choices each time!
When choosing r of them, the permutations are:
n × n × ... (r times)
(In other words, there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on, multplying each time.)
Which is easier to write down using an POWER of r:
n × n × ... (r times) = nr
Example: in the lock above, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them:
10 × 10 × ... (3 times) = 103 = 1,000 permutations
So, the formula is simply:
| nr |
| where n is the number of things to choose from, and you choose r of them (Repetition allowed, order matters) |
2. Permutations without Repetition
In this case, you have to reduce the number of available choices each time.
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For example, what order could 16 pool balls be in?
After choosing, say, number "14" you can't choose it again.
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So, your first choice would have 16 possibilites, and your next choice would then have 15 possibilities, then 14, 13, etc. And the total permutations would be:
16 × 15 × 14 × 13 × ... = 20,922,789,888,000
But maybe you don't want to choose them all, just 3 of them, so that would be only:
16 × 15 × 14 = 3,360
In other words, there are 3,360 different ways that 3 pool balls could be selected out of 16 balls.
But how do we write that mathematically? Answer: we use the "factorial function"
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The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples:
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| Note: it is generally agreed that 0! = 1. It may seem funny that multiplying no numbers together gets you 1, but it helps simplify a lot of equations. | |
So, if you wanted to select all of the billiard balls the permutations would be:
16! = 20,922,789,888,000
But if you wanted to select just 3, then you have to stop the multiplying after 14. How do you do that? There is a neat trick ... you divide by 13! ...
16 × 15 × 14 × 13 × 12 ...
| = 16 × 15 × 14 = 3,360 | |
13 × 12 ...
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Do you see? 16! / 13! = 16 × 15 × 14
The formula is written:
 |
| where n is the number of things to choose from, and you choose r of them (No repetition, order matters) |
Examples:
Our "order of 3 out of 16 pool balls example" would be:
| 16! | = | 16! | = | 20,922,789,888,000 | = 3,360 |
| (16-3)! | 13! | 6,227,020,800 |
(which is just the same as: 16 × 15 × 14 = 3,360)
How many ways can first and second place be awarded to 10 people?
| 10! | = | 10! | = | 3,628,800 | = 90 |
| (10-2)! | 8! | 40,320 |
(which is just the same as: 10 × 9 = 90)
Notation
Instead of writing the whole formula, people use different notations such as these:
