Mathematical Permutations

Introduction :

In this article, we are going to discuss about the concepts of the mathematical permutations. The mathematical term “permutation” is nothing but “arrangement”. For instance, given 3 symbols x, y, z suppose we arrange them taking 2 at a time.

The various arrangements are xy, yx, yz, zy, xz, zx.

Therefore, the total number of arrangements of 3 things taken 2 at a time is 6 and it can be written as `^3P_2 ` = 6. Now we are going to see some example problems based on the mathematical permutations.
Mathematical Permutations :

Definitions:

Taking r at a time from the arrangements made out for n things is referred to as number of permutations of n things taken r at a time.

`^n P _r = (n!)/((n-r)!)` .

`^n P _n = n!` .

`0! = 1! = 1` .

Notation:

In permutation, n and r are the two real positive numbers with r is greater than or equal to 1 and less than or equal to n, ie. (`1<= r <= n` ). Then the numbers of all permutations of n individual items are taken to denote r at a time, It expressed in symbol as P (n, r) or `^nP_r` . We use the illustration `^nP_r` each and every one of the way through our discussion. Therefore the value of `^nP_r` is given as the total number of permutations of n distinct things taken r at a time.

 

 

 

Note:

Only when the arrangement of order is changed we can obtain different types of permutation. These arrangements of orders are included to the account in permutation.
Examples for Mathematical Permutations :

Example 1:

Determine how many arrangements of the symbols 1, 2, 3, 4 and taking three at a time.

Solution:

Given:

Number of symbols in a set , n = 4.

Number of symbols to pick from the set, r = 3.

All the permutations. `^n P _r = (n!)/((n-r)!)` .

`= (4!)/((4-3)!)` .

`= (4!)/((1)!)`.

`= (4 xx 3 xx 2 xx 1)`.

`= 24`.

Therefore, the required permutations are shown below,

{1,2,3} {1,2,4} {1,3,2} {1,3,4} {1,4,2} {1,4,3} {2,1,3} {2,1,4} {2,3,1} {2,3,4} {2,4,1} {2,4,3} {3,1,2} {3,1,4} {3,2,1} {3,2,4} {3,4,1} {3,4,2} {4,1,2} {4,1,3} {4,2,1} {4,2,3} {4,3,1} {4,3,2}.
Example 2:

Evaluate: `^7 P _3` .

Solution:

Given:

`^7 P _3` .

To find the permutation of the given expressions;

Here, n = 7, and r = 3.

Formula:

According to the permutaion formula. `^n P _r = (n!)/((n-r)!)` .

`= (7!)/((7-3)!)` .

`= (7!)/((4)!)`.

`= (7xx 6 xx 5 xx4!)/(4!)`.

`= (7xx 6 xx 5)`.

`= 210`.