Questions tagged [permutations]

For questions related to permutations, which can be viewed as re-ordering a collection of objects.

The word permutation has several possible meanings, based on context. In combinatorics, a permutation is generally taken to be a sequence containing every element from a finite set exactly once. Permutations of a finite set can be thought of as exactly the ways in which the elements of the set can be ordered.

In group theory, a permutation of a (not necessarily finite) set $S$ is a bijection $\sigma : S \to S$. The set of all permutations of $S$ forms a group under composition, called the symmetric group on $S$.

Reference: Permutation.

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Multiplication in Permutation

I have a very very stupid question... But i can't get the reason. Why do we use multiplication in permutations? For example, Lets say we have 5 friends, A,B,C,D,E. How many ways can we pick a Gold, Silver, and Bronze medal for “Best friend in…
Rebooting
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Cycle structure in a symmetric group

I have a bit of a problem. I'm currently reading about permutations, and I have a little exercise that asked me to find all cycle structures in $S_6$. I came up with the following $ ( -)\\ (- -)\\ (- - -)\\ (- - - -)\\ (- - - - -)\\ (- - - - -…
JustDanyul
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How to determine the parity of a permutation by its cycle decomposition

If one is given the length of a permutation and the number of cycles, is it possible to determine the parity of the permutation? Oddly enough, there's no definition in the text I'm reading for cycle decomposition. I'm assuming it means the…
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What is a simple proof for a $k-\text{Cycle}$ Permutation needing at least $(k-1)$ transpositions in its decomposition?

I am a high school teacher attempting to prove in a programming class why the answer to http://www.usaco.org/index.php?page=viewproblem2&cpid=785, at least if all the cow heights are unique, is $1$ less than the number of cows that are out of order…
jdowdell
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How does a permutation act on a string?

Is there a conventional way to have a permutation act on a list of objects? It seems like there are two possible ways, one being the inverse of the other. Suppose I have a permutation $\sigma \in S_4$ which is concretely specified as a function…
Paul Orland
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Explain method behind permutations question (high school level)

I am in high school and got a question in my textbook which reads: A baby has nine different toy animals. 5 of them are red and 4 of them are blue. She arranges the toys in a line so that the colours are arranged symmetrically. How many…
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Why do we divide values to count permutations with repeated letters?

My question is why we divide values in permutations in case of repetition of words Like if we have to arrange the word "coffee" in different ways then my answer will be $6! \over {2!\times 2!}$ (i know the complete formula). I can't understand why…
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How many integers between 1000 and 9999 inclusive consist of

How many integers between $1000$ and $9999$ inclusive consist of (a) Distinct odd digits, (b) Distinct digits, (c) From the number of integers obtained in (b), how many are odd integers? (a) ${}^5P_4 = 120$ (b) $9 \times 9 \times 8 \times 7 =…
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|x|+|y|+|z|=15. How many integer solutions do exist?

I tried to truncate the problem to $|x|+|y|=15$, which is giving me the answer to be $60$ solutions. However, I am trying to apply beggar method and everything shatters. How should I proceed? Can I apply beggar method here? Is the answer somewhat…
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Number of ways to order a list of permutations by swapping

I want to solve a problem and I have no idea how or where to begin. I don't even know if it's possible to solve. I tried to find any clues in books about discrete maths but I didn't find anything that could help me solve the problem. Suppose there…
sclaes
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Does the set of permutations of an empty set contain an empty set?

So the set of permutations for $\{x, y\}$ is $\{(x, y), (y, x)\}$. However, if I would try to make a set of permutations of an empty set $∅$, would the permutation set be $∅$ or $\{∅\}$?
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The parity of the permutation $a \mapsto ma \bmod{n}$

I am working on a unique kind of permutations, and would like to know if there is a quick way to know what is the parity of each of them. Given an integer $n$, I can take any integer $m$ for which $\mathrm{gcd}(m,n)=1$, and then define for each $0…
IBS
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A family consisting of 2 parents and 3 children is to pose a picture with 2 family members in the front and 3 in the back.

a)How many arrangements are possible with no restrictions? b) How many arrangements are possible if the parents must sit in the front? c) How many arrangements are possible if parents must be next to each other? My answers are a) 5! …
R.Temur
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Find the sum of all 4-digit numbers formed by using the digits 1, 2, 3, 4, 5

...and where no digit is to be repeated in any 4-digit number. Yes, I am aware of similar questions on this site, but none of the answers gave me insight as to how this works. I would like an answer in ridiculously simple terms, like explaining it…
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Find the word at $48$ position?

The letters in the word "PLACES" are permuted in all possible ways and arranged in the alphabetical order.Find the word at 48 position. a)AESPCL b)ALCEPS c)ALSCEP d)AESPLC MyApproach As per dictionary I started with…
justin takro
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