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.

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Permutations of 3 digit numbers divisible by 5

I recently had to answer the following permutation question: How many 3 digit numbers can be formed from the digits 2,3,5,6,7,9 which are divisible by 5 and none of the digits are repeated? Having not done this kind of algebra since high school…
Ben Foster
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In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A?

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A? I have a doubt in the selection of A at the last position. Please help. Thanks in advance!!
user2689798
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Need help calculating number of possible passwords with given criteria

I need help calculating the number of possible passwords with a given set of criteria. Here is the set of criteria: Passwords are case insensitive. Must be 6-14 characters. Must contain at least 1 letter and 1 number. Must not be equivalent to your…
dprofancik
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Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$.

$S_n$ is the set of all permutations. Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$ I can somewhat see why this statement would be true, seeing as permutations are read from right to left which could have something to do with the order…
Chris
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Different ways in which a micro-switch with eight switches can set

A computer interface for a Kawai digital studio piano has eight micro-switches that can be set in either the "on" or "off" position. These switches must be set properly for the interface to work. In how many different ways can this group of switches…
Ben
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Series that converge to every real number via permutation

This great answer at MathOverflow, https://mathoverflow.net/a/29488/8784, shows that the set of permutations of $\mathbb N$ is uncountable. However, I did not grasp the fact that he uses: any conditionally convergent series [and that such exists]…
Mohammad Al-Turkistany
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What happens to the Permutation Rule when r=0?

This is small but quirky idea that popped into my head in the middle of the night last night. If I have $n$ objects, and want to find out how many permutations (sequences) of $r$ objects there are, we use: $_nP_r = \frac{n!}{(n-r)!}$. Say for…
Monica Heddneck
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circular permutation in case of identical diamond and pearls

the number of ways a necklace can be formed by 18 identical diamond and 3 identical pearl ? my solution is that divide the procedure into three case case1:all pearls together that leads to only one case. case 2: two pearls together and one…
kola
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Counting powers of permutations

I didn't find similar questions so decided to ask this one. Given positive integers $n$ and $d$ how can we efficiently estimate (or better calculate) cardinality of the set $~~ \{ \sigma^d ~~|~~ \sigma \in S_n \} ~~$? Here $S_n$ denotes the set of…
Igor
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How many words, with or without meaning, can be formed by selecting $3$ consonants and $2$ vowels from $7$ consonants and $4$ vowels?

There are $7$ consonants and $4$ vowels. How many words, with or without meaning, can be formed by selecting $3$ consonants and $2$ vowels? Should one consider permutation or combination?
Achari S Ganesha
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Circular permutation - Arranging 4 persons around a circular table where 8 seats are there. (cond.)

Suppose 4 persons A,B,C and D sit around a round table with 8 seats. Rotation by 8,16,24,... seats defines same arrangement and other rotations gives different arrangements. If seats are identical, there are 7*6*5 arrangements as clarified…
Kiran
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How to solve this permutation math problem?

In how many ways can 4 girls and 2 boys sit at a movie theater row with 6 seats if a girl must be seated at each end.
Adam
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Permutation multiplication of non-disjoint cycles in $S_4$

I am having an issue with calculating the product of permutation cycles for calculating commutators. Consider $A_4$ and the commutator $[(123),(14)(23)]$ So we have $[(123),(14)(23)] = (132)(14)(23)(123)(14)(23)$ I think am calculating incorrectly…
oliverjones
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Is the cyclic permutation $(1 2 3)$ equal to $(12)$ followed by $(13)$ or $(13)$ followed by $(12)$

At the bottom of this page: http://dogschool.tripod.com/permgroups.html it states that (1 2 3) This is equivalent to two transpositions: (1 2) followed by (1 3) [try it!] So I did try it: ( 1 2 3 ) means that 1 goes to 2 which goes to 3 which…
Augs
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Can someone explain this proof that each transposition changes the parity of a permutation?

Looking at: http://dogschool.tripod.com/permutation.html It states: Let the following represent an ordering of the n numbers 1 to n. a1 a2 . . . x . . . y . . . an-1 a1 Each number in the string of numbers can affect the grand total in two ways.…
Augs
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