Permutation is one of the subjects discussed in the discipline of combinatorial analysis in math. Having in hand any ordered sequence with an “n” number of distinct elements, any other sequence formed by the same “n” reordered elements is called a permutation.
Thus, we can say that if A is a permutation of B, then A and B are made up of the same elements, but ordered differently.
Where do permutations come from?
Permutations are isolated cases of Simple Arrangements. These are ordered groupings of a set A of elements, such that the groups have fewer or equal numbers of elements than set A.
The set A = {X, Y, Z}, {X, Y} and {Y, X} is a simple arrangement of the elements from A taken 2 to 2. The number of elements in A is represented by the letter “n”. O order number, or class number, is “k”. This number is the number of elements in each simple array (in the case of the example, this number is 2).
The list with all the simple arrangements of the three elements of A taken 3 to 3 is as follows:
XYZ, XZY, ZXY, ZYX, YZX and YXZ
This list is precisely the particular case of arrangements that receive the name of permutation.
Calculation of simple arrangements
The number of simple arrangements of a set A, which has no elements taken k The oh, can be calculated by the following formula:
THEno, ok = no!
(n - k)!
Permutation Definition
Let A be a set with no distinct elements. You simple arrangements of these elements taken n to n are called simple permutations of A. Thus, for it to be a permutation, it is necessary that the order number k be equal to the number no of elements of A. The following calculation results from this:
Taking the formula used for simple arrays and the order number k = n, we will have:
This is the formula used to calculate the number of permutations of the elements of the set A, usually denoted by Pno. Soon:
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Pno = Ano, no = n!
Pno = n!
Example
Calculate the number of permutations of the letters of the word LOVE.
Solution:
Note that the word LOVE has 4 distinct elements. To calculate the number of permutations of this word, we will use the formula above:
Pno = n!
P4 = 4!
P4 = 4·3·2·1
P4 = 24
Therefore, it is possible to form 24 different permutations of the letters of the word LOVE. Word permutations are also called anagrams.
Permutations with repeated elements
Any set can have repeated elements. At permutations that set should consider the repetition of these elements, because the order in which they appear does not matter, unlike the order of the other elements of the set. If we change only the two “A”s of place in the word AMAR, we will get the same word. Alike words are not permutations, therefore, this repetition must be subtracted in the formula for the permutations.
To subtract all possible repetitions of elements in one permutation with repeated elements, we must do the following:
Let A be a set with no elements, of which k elements repeat themselves. The formula for calculating the permutations of A is:
Pnok = no!
k!
If set A, with no elements, possess k repetitions of an element and j repetitions of another, the calculation will happen as follows:
Pnohaha = no!
k!·j!
If a set A, with no elements, has k repetitions of an element, j repetitions of another, …, m repetitions of another, the formula takes the following form:
Pnok, j,...,m = no!
k!·j!·... ·m!
Example
Calculate the number of anagrams of the word ANTONIA.
Solution:
To solve the example, just calculate the permutations with repeated elements of the word ANTONIA. Both the letter A and the letter N are repeated 2 times. Watch:
P72,2 = 7!
2!·2!
P72,2 = 7·6·5·4·3·2·1
2·1·2·1
P72,2 = 5040
4
P72,2 = 1260
By Luiz Paulo Moreira
Graduated in Mathematics
