Mathematics Stack Exchange
Mathematics Stack Exchange

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.

12854 questions
4
votes
1 answer

How to intuitively understand the formula for permutation on non-distinct objects?

I have clear understanding of permutation but I still can not intuitively understand why in permutation for non-distinct elements we divide number of possible arrangements by the factor of number of non-distinct elements. $n!/(n_1! \times n_2!…
R.Temur
  • 415
4
votes
4 answers

Determine the number of permutations of $ \ \{1,2,3,4,5,6,7,8,9,10\} \ $ that have exactly 3 numbers in their natural position

Determine the number of permutations of $ \ \{1,2,3,4,5,6,7,8,9,10\} \ $ that have exactly 3 numbers in their natural position. $$ $$ Is it $ \ \ \begin{pmatrix} 10 \\ 3 \end{pmatrix} \times 7 ! \ $ ?
MAS
  • 10,638
4
votes
2 answers

Permutations : A person picks 4 numbers from a set of 9, what are the total ways he can Win?

In a casino, a person picks 4 numbers from a set of 9 numbers (3 Even, 6 Odd). A person wins if at least one of those 4 numbers is Even. What are the total ways he can Win ? There are two approches I believe. 1) One is to find all possible ways of…
Raghu
  • 47
4
votes
2 answers

Sitting arrangement problem.

How many ways are there to seat down 5 boys and 5 girls, on two parallel benches of length $5$, such that there is at least one girl opposite in front of a boy? My attempt - I tried to solve it by finding total ways and subtract the case when no…
user404716
4
votes
2 answers

How many points would be required to win a football group with 6 teams?

Suppose you have a group of 6 football teams. The same as the World Cup, but with 6 teams instead of 4. Suppose each team plays each other twice, meaning each team plays 10 games. A win grants 3 points, a draw grants 1 point and a loss grants 0…
Max Jarl
  • 41
4
votes
1 answer

Number of permutations with a fixed point

I "created" this excercise on my own when I worked on a task relating to stochastics, but I might need a little bit of help. Task Let $M = \{1, 2, ..., N\}$ be a set and $N!$ the number of permuations $\sigma$ on $M$. Furthermore, let $k$ be the…
Julian
  • 1,401
4
votes
2 answers

permutation with repetition subset

How many distinct strings of length 4 can be generated with $c,b,b,a,a,d$ Through a script I know that there are 102 such possibilities. My Attempt Case 1: using only one 'b' and one 'a'. This can happen in 4! number of strings. Case 2: using two…
Sohail
  • 145
4
votes
2 answers

Permutations and Combinations Heads/Tails

Out of all of the potential sequences of 73 Heads/Tails games, each being Heads or Tails, how many sequences contain 37 tails and 36 heads? Express the output in terms of factorials. Because there are sequences of 73 games my initial thought is,…
MathsUndergrad
  • 195
4
votes
1 answer

Find the element with the highest order in a symmetric group?

How does one find the element with the highest order in $S_9$ ? I can only guess that the element may be (1 2 3 4)(5 6 7 8 9) whose order is 20.
Rescy_
  • 2,002
4
votes
3 answers

How many 3 digit numbers can be formed using digits 1,2,3,4 and 5 such that the number is divisible by 6

How many $3$ digit numbers can be formed using digits $1,2,3,4$ and $5$ without repetition such that the number is divisible by $6$ First Approach: A number is divisible by $6$ if it is divisible by $2$ and $3$. Now the possible combinations I…
justin takro
  • 1,288
4
votes
5 answers

Find the sum of all 4 digit numbers which are formed by the digits 1,2,5,6?

I have researched and found 2 approaches but haven't understood both.Can anyone explain it clearly or probably with any real world example? Approach 1 Four digit numbers $ = 4 \cdot 3 \cdot 2 \cdot 1 = 24$ ways we can form a four digit number.…
Jack
  • 752
4
votes
3 answers

Permutation of Numbers

How many $5$ digit numbers can be formed from the integers $\{1,2,...,9\}$ if no digit can appear more than twice.(for example 41434 is not allowed) My approach is : Since, $max $ 2$ digits can repeat: $=9\times9\times8\times7\times6$ Total Numbers…
Rahul
  • 73
4
votes
2 answers

$n$-permutations with exactly $k$ fixed points

It's easy to deduce the formula for $n$-permutations with exactly $k$ fixed points. The result is similar to $n$-derangement formula and it's equal to $ D_{n,k}= \frac{n!}{k!}\sum_{i=0}^{n-k}\frac{(-1)^i}{i!}$. But I think it is not convenient. Last…
xan
  • 1,443
  • 1
  • 21
  • 28
4
votes
1 answer

Which other "exotic" permutation-related things exist?

Some time back I posted some questions about the "exotic" outer automorphisms of $S_6$, and part of the answer was a citation of a paper by T. Y. Lam that said, among other things, that the group of all automorphisms of $S_6$ has exactly three…
Michael Hardy
  • 1
4
votes
1 answer

Permutation questions

What is the smallest number n such that $A_{n}$ contains a permutation of order 2004? I calculated it to 334, but the answer is 176 and I can't see why? I first noticed that $2004 = 2^{2}\cdot 3 \cdot 167$ then I looked at the elements in $A_{n}$…
bemyguest
  • 567
1 2 3
…
69 70