In Mathematics, more precisely in the contents of combinatorial analysis, permutations between the letters of a word, between the numbers of a sequence, between the elements of a set, and so on are called anagrams.
In this way, the calculations involving anagrams they will usually aim to find out how many ways it is possible to reorder the elements of a set in which the order of those elements matters. For example: in how many ways is it possible to choose the password for a credit card, knowing that four digits from 0 to 9 can be chosen without repeating any digits?
What is permutation?
Permutation it is the exchange of place between two or more elements of an ordered list or set. O Fundamental Principle of Counting allows the permutations between these elements to be counted. Of course, it is often not possible to count these exchanges in the literal sense of the word. However, they can be calculated by the aforementioned principle.
As a anagram is a new word or list obtained through the elements of another word or list, so it is obtained with a permutation.
anagram examples
The word OVA has the following anagrams:
OVA, OAV, VOA, VOA, AOV and AVO
Some of the anagrams of the word PATO are:
DUCK, TOPA and OPTA
Anagram Calculation
First, when the anagrams are of words that have all the different letters, the possibility of choosing letters for the first space of the new word is the total number of letters (n). For the second space, the letter chosen in the first space cannot be repeated, so the amount of choice for that space is “n – 1” and so on. Watch:
Example: How many anagrams are there in the word TOPA?
Note that the word "TOPA" has no repetition of letters, so we will use the fundamental principle of counting, or simple permutation:
4·3·2·1 = 24
The word "TOPA" itself is already included in this result, so the number of anagrams for that word is 24 - 1 = 23.
On the other hand, there are cases where anagrams of words that have repeating letters. Follow the development of one of these cases in the following example:
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Example: How many anagrams are there in the word PINEAPPLE?
There are 5 letters available for exchange in 7 spaces. Note that the letter A repeats 3 times. To consider this repetition when calculating the quantity of anagrams, follow the reasoning: If the letter A is used in the first space, it can still be used in the second. Therefore, it is still possible to choose five different letters for the second space.
Assuming it is also used in the second, there are still five different letters left for the third. Finally, if it is used in the third, it is no longer possible to have the letter A and therefore only 4 different letters are left for the fourth. The calculation to be done will be as follows: calculate the permutation of 7 letters and divide the result by the "permutation" of the repeating letters:
7! = 7·6·5·4·3·2·1 = 5040 = 840
3! 3·2·1 6
So there are 840 anagrams with the word PINEAPPLE.
This is also the way to proceed when the word to calculate the amount of anagrams features more than one repeated letter. Note the following example:
Example: Calculate the number of anagrams of the word MOM, disregarding the accent.
There are three different letters for 5 spaces, with a repetition of the letter M and one of the letter A. In the first two spaces, we will have 3 possibilities of letters, in the next two, only two possibilities, and for the last space, we will have only one possibility. By dividing the permutation of 5 "spaces" by the permutations of the repeating letters, we will have:
5! = 120 = 120 = 30
2!2! 2·2 4
There are 30 - 1 = 29 anagrams of the word MOM, disregarding the accent.
By Luiz Paulo Moreira
Graduated in Mathematics
Would you like to reference this text in a school or academic work? Look:
SILVA, Luiz Paulo Moreira. "What is an anagram?"; Brazil School. Available in: https://brasilescola.uol.com.br/o-que-e/matematica/o-que-e-anagrama.htm. Accessed on June 27, 2021.
