Permutation definition and examples

This article will be about What is the permutation, how the use of it, with solved examples and the formula.

¿What is the permutation?

The permutation is a mathematical method used in statistic where we can define of how many different ways we can select some elements from a set. for example, if we have a set with 20 elements, the permutation would allows us to find the number of ways we can select a determined number of elements.

The repetition of elements is not something allowed in the permutation, this means that an element cannot be selected twice, something we can do with the combination.

Permutation

Something that we have to take into account is that the order of the elements is important, for example, if we have to order 3 elements, in the permutation is not only taken into account that 3 of the elements coincide, it will matter the order in which they were selected, where each selection is a different permutation.

As we said in the definition of permutation, this is useful to define of how many ways we can classify or order a set of elements in a smaller set formed by elements of the major set, when we talk about a smaller set we are referring to extract elements from the set to form another one.

Basic example of permutation

To understand this a little bit better we are going to develop the following example: if we have a bucket with 4 balls that are named by the letters “a, b, c and d”, and we pull out 2 balls, not at the same time but one first and then the other, ¿Of how many ways we can pull out these two balls?

The following table represent every possible way to do it.

N° Ball 1 Ball 2
1 A B
2 A C
3 A D
4 B A
5 B C
6 B D
7 C A
8 C B
9 C D
10 D A
11 D B
12 D C

Pay attention on how the permutation 1 and 4 are different even when the balls are the same, this is because in the permutation the order does matter and not only the combination of balls.

In total there are 12 different ways to pull out the balls, this may look simple by calculating it using the table or with a tree diagram, but the bigger the set is the harder it is to calculate it, this is why we use the permutation formula when it comes to big groups of numbers or things.

Formula for the permutation

The permutation formula uses factorials, as a reminder, a factorial is the number multiplied by every integer number below it, for example the factorial of 4 (4!) is 4x3x2x1, and we also know that the factorial of 0 is 1.

The permutation is written this way: nPr, where “n” is the total of elements, and “r” is the subset of elements we want to order, and “r” can not be bigger than “n”.

  • Formula
  • nPr =
    n!/(n-r)!

Examples of permutation

Example 1 There is a competition between 5 football teams, if they are going to give a price for the 1st, 2nd and 3rd place, of how many ways this places can be ordered.

n = 5 r = 3 5P3 = ?

  • nPr =
    n!/(n-r)!
  • 5P3 =
    5!/(5-3)!
  • 5P3 =
    5!/2!
  • 5P3 =
    5*4*3*2*1/2*1
  • 5P3 =
    120/2
  • 5P3 = 60

Example 2 If we choose the winners of the first and second price of a giveaway that was made on a group of 10 people, how many winner combinations could there be.

n = 10 r = 2 10P2 = ?

  • nPr =
    n!/(n-r)!
  • 10P2 =
    10!/(10-2)!
  • 10P2 =
    10!/8!
  • 10P2 =
    10*9*8*7*6.../8*7*6*5*4...
  • 10P2 =
    3 628 800/40 320
  • 10P2 = 90

Example 3 In a hospital there are 4 chairs where 4 patients can seat, if the order of the people on the chairs define their turn, of how many ways can the patients be ordered.

n = 4 r = 4 4P4 = ?

  • nPr =
    n!/(n-r)!
  • 4P4 =
    4!/(4-4)!
  • 4P4 =
    4!/0!
  • 4P4 =
    4*3*2*1/1
  • 4P4 =
    24/1
  • 4P4 = 24

Example 4 If there is a container where there are 12 papers, each one with a different letter, and we took out 5 papers one by one ¿of how many ways can we took out these papers?

n = 12 r = 5 12P5 = ?

  • nPr =
    n!/(n-r)!
  • 12P5 =
    12!/(12-5)!
  • 12P5 =
    12!/7!
  • 12P5 =
    12*11*10*9*8.../7*6*5*4*3...
  • 12P5 =
    479 001 600/5040
  • 12P5 = 95 040

Example 5 In a classroom there are 8 students and they will give a medal to the 1st 2nd and 3rd best student, calculate every combination.

n = 8 r = 3 8P3 = ?

  • nPr =
    n!/(n-r)!
  • 8P3 =
    8!/(8-3)!
  • 8P3 =
    8!/5!
  • 8P3 =
    8*7*6*5*4.../5*4*3*2*1
  • 8P3 =
    40 320/120
  • 8P3 = 336

Example 6 A software generates a password automatically every time we press a button, where the password is 5 characters long, and there are 20 possible characters (the characters can not be repeated). How many different password can the software generate.

n = 20 r = 5 20P5 = ?

  • nPr =
    n!/(n-r)!
  • 20P5 =
    20!/(20-5)!
  • 20P5 =
    20!/15!
  • 20P5 =
    20*19*18*17*16.../15*14*13*12*11
  • 20P5 =
    2 432 902 008 176 640 000/1 307 674 368 000
  • 20P5 = 1 860 480

Example 7 A directive will be chosen on a work group where there will be a president, vice president and secretary, if there are 10 people in this group, how many different directives can they form.

n = 10 r = 3 10P3 = ?

  • nPr =
    n!/(n-r)!
  • 10P3 =
    10!/(10-3)!
  • 10P3 =
    10!/7!
  • 10P3 =
    10*9*8*7*6.../7*6*5*4*3...
  • 10P3 =
    3 628 800/5040
  • 10P3 = 720

Example 8 8 runners participate in a race where there will be a price for the first 5 runners (each place will have a better price) how many ways we can price the runners.

n = 8 r = 5 8P5 = ?

  • nPr =
    n!/(n-r)!
  • 8P5 =
    8!/(8-5)!
  • 8P5 =
    8!/3!
  • 8P5 =
    8*7*6*5*4.../3*2*1.
  • 8P5 =
    40320/6
  • 8P5 = 6720

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Combination Classical probability Empirical probability Subjective probability Sum rule Multiplication rule Complement rule Bayes theorem Total probability