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Basic tutorials Permutation
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Permutations
There are basically two types of permutation:
1) Repetition is Allowed: such as the lock above. It could be "333".
2) 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!
So when choosing r of them, the permutations are:
n x n x ... (r times) = nr
(Because there are n possibilities for the first choice, THEN there are n possibilites for the second choice, and so on.)
Example: in the lock above, there are 10 numbers to choose from (0,1,..9) and you choose 3 of them:
10 x 10 x ... (3 times) = 103 = 1000 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.
| For example, what order could 16 pool balls be in? |
| After choosing, say, number "14" you can't choose it again |
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 x 15 x 14 x 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 x 15 x 14 = 3360
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"
The factorial function (symbol: !) just means to multiply a series of descending natural numbers. Examples:
1 4! = 4 × 3 × 2 × 1 = 24
2 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040
3 1! = 1
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 x 15 x 14 x 13 x 12 …./13 x 12) = 16 x 15 x 14 = 3360
Do you see? 16! / 13! = 16 x 15 x 14
The formula is written:
(n!/(n-r)!)
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!/(16-3)!} = (16!/13!) = (20,922,789,888,000 / 6,227,020,800) = 3360
(which is just the same as: 16 x 15 x 14 = 3360)
How many ways can first and second place be awarded to 10 people?
{10!/(10-2)!} = (10!/8!) = (3,628,800/40,320_ = 90
(which is just the same as: 10 x 9 = 90)
Notation
Instead of writing the whole formula, people use different notations such as these:
Example: P(10,2) = 90
Permutation and Combination Formula & Tips by Sandeep Jain, S.S. Education
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