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Permutations and combinations

Introduction..... Symbols..... Permutations..... Permutations with repetition..... Combinations..... Combinations with repetition.....



Introduction

A permutation is an arrangement of things with the order being important. abcd is different to bcda.  A combination is a selection of things e.g. abcd is the collection of the four letters with no relevance given to the order.   If a team of eleven players are selected for a football team out of a selection of 20 persons with no importance given to the positions then the selection is combination.  However if each position (goalkeeper, left fullback etc. ) is chosen seperately then the selection is a permutation.




Symbols..
n P(r)  or   n P (r) = Permutation of n items taken r at a time
n C(r)  or   n C (r) = Combination of n items taken r at a time




Permutations and combinations



Permutations: (Without Repitition)

Permutation illustrates the number of ways to arrange elements in a definite order.
The number of permutations of n items taken all at a time =

n! = 1.2.3 .....n     (N Factorial)

The number of permutations of n elements of same kind taken r at a time.

nP(r) = n ! /(n-r)!

Example
The number of permutations of four l etters a,b,c,d arranged three at a time is
4! /(4-3)! = 4.3.2.1 / 1 = 24

a,b,c / a,b,d /a,c,b /a,c,d / a,d,b/ a,d,c
b,a,c / b,a,d /b,c,a /b,c,d / b,d,a /b,d,c
c,a,b /c,a,d / c,b,a / c,b,d / c,d,a / c,d,b
d,a,b / d,a,c / d,b,a/ d,b,c / d,c,a /d,c,b

Permutation Table (without Repetition)
n r
1 2 3 4 5 6 7 8 9 10
1 1 0 0 0 0 0 0 0 0 0
2 2 2 0 0 0 0 0 0 0 0
3 3 6 6 0 0 0 0 0 0 0
4 4 12 24 24 0 0 0 0 0 0
5 5 20 60 120 120 0 0 0 0 0
6 6 30 120 360 720 720 0 0 0 0
7 7 42 210 840 2520 5040 5040 0 0 0
8 8 56 336 1680 6720 20160 40320 40320 0 0
9 9 72 504 3024 15120 60480 181440 362880 362880 0
10 10 90 720 5040 30240 151200 604800 1814400 3628800 3628800




Permutations: (With Repetition)

A Permutation with repetition is the number of ways to arrange elements in a definite order allowing elements to be repeated.
The number of permutations with repetition of n items taken r at a time =

nP(r) (with repetition) = n r

Example
The number of permutations of three letters a,b,c arranged two at a time is
3 2 = 9

a,a / a,b / a,c / b,b /b,a / b,c/ c,c /c,a / c,b

Permutation Table (With repetition)
n r
1 2 3 4 5 6 7 8 9 10
1 1 0 0 0 0 0 0 0 0 0
2 2 4 0 0 0 0 0 0 0 0
3 3 9 27 0 0 0 0 0 0 0
4 4 16 64 256 0 0 0 0 0 0
5 5 25 125 625 3125 0 0 0 0 0
6 6 36 216 1296 7776 46656 0 0 0 0
7 7 49 343 2401 16807 117649 823543 0 0 0
8 8 64 512 4096 32768 262144 2097152 16777216 0 0
9 9 81 729 6561 59049 531441 4782969 43046721 387420489 0
10 10 100 1000 10000 105 106 107 108 109 1010




Combinations:

A combination is the number of ways to arrange elements with no definite order.  The number of combinations of n elements of same kind taken r at a time.


Equals
n C(r) = n ! / r !(n-r)!

Example.
The number of combinations of four letters a,b,c,d arrange three at a time is
4! / 3!(4-3)! = 4.3.2.1 / 6.1 = 4

a,b,c / a,b,d /a,c,d /b,c,d



Combination Table (without Repetition)
n r
1 2 3 4 5 6 7 8 9 10
1 1 0 0 0 0 0 0 0 0 0
2 2 1 0 0 0 0 0 0 0 0
3 3 3 1 0 0 0 0 0 0 0
4 4 6 4 1 0 0 0 0 0 0
5 5 10 10 5 1 0 0 0 0 0
6 6 15 20 15 6 1 0 0 0 0
7 7 21 35 35 21 7 1 0 0 0
8 8 28 56 70 56 28 8 1 0 0
9 9 36 84 126 126 84 36 9 1 0
10 10 45 120 210 252 210 120 45 10 1




Combinations: with repetition

Combinations with repetition are the number of ways to arrange elements with no definite order but the elements can be repeated.   The number of combinations of n elements of same kind taken r at a time.


Equals
n C(r) = (n + r -1)! / r ! (n - 1)!

Example.
The number of combinations of four letters a,b,c arrange two at a time is
(3 + 2 - 1)! / (2)!.(3-1)! = 4.3.2.1 / (2.1)(2.1) = 6

a,a /a,b / a,c / b,b / b,c/ c,c /

Combination Table (with repetition)
n r
1 2 3 4 5 6 7 8 9 10
1 1 0 0 0 0 0 0 0 0 0
2 2 3 0 0 0 0 0 0 0 0
3 3 6 10 0 0 0 0 0 0 0
4 4 10 20 35 0 0 0 0 0 0
5 5 15 35 70 126 0 0 0 0 0
6 6 21 56 126 252 462 0 0 0 0
7 7 28 84 210 462 924 1716 0 0 0
8 8 36 120 330 792 1716 3432 6435 0 0
9 9 45 165 495 1287 3003 6435 12870 24310 0
10 10 55 220 715 2002 5005 11440 24310 48620 92378


Useful Related Links
  1. tutors4You - Permutations and Combinations ...Clear tutorials on permutations and combinations
  2. TheMathPage Permutations and Combinations.. Basic study of the topic.
  3. Statistics Glossary .... Very accessible notes with some detail.

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Last Updated 21/09/2007