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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How many ways can 7 people be seated at a round table?

In how many ways can 7 people be seated at a round table if they can sit anywhere? The answer is 6!=720 I expected 7! I don't understand why. Does it have anything to do with the fact that the table is round?
TSR
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How many 5 digit numbers can be made by using 0~9, that is bigger than 12345?

So using numbers from 0~9, making a 5 digit number, how many numbers can be formed that is bigger than 12345? Repetition is not allowed. Thank you.
Sadpersonn
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Order of permutation $a \in S_n$ if $a^k$ is a cycle of length n

Let $a$ be an element of $S_n$ ,the permutation group of order n. $a^k$ is a cycle of length n. Then what is the order of $a$? If $n$ is prime then a should be a cycle of length $n$.But if $n$ is not prime then how to find the order of $a$? Let…
Unknown
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Average distance moved by a permutation

For each $\pi$ permutation of numbers 1 to n, $f(\pi) = \sum_{i=1}^n |\pi_i - i|$. What is the average of $f(\pi)$ on all of the $n=7$ permutations?
Developer
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Sending permutations

This seems simple but has stumped me for around half an hour... The paragraph reads: "Let $x_1,x_2,x_3$ be three variables. We let the permutations in $S_3$ move around like $1$, $2$, $3$". So for instance, the permutation $(132)$ sends $$…
user635953
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factorial of no. which are is in the form of $5m+1,5m+3$

[a] The no. of positive integer divisers of $10!$ which are is in the form of $5m+1\; \forall m\in \mathbb{N}$ [b] The no. of positive integer divisers of $10!$ which are is in the form of $5m+2\; \forall m\in \mathbb{N}$ [c] The no. of positive…
juantheron
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$S_n$ embeds into $A_{n+2}$.

I was reading this post explaining why $S_n$ embeds into $A_{n+2}$. So they suggest that the embedding schould be $Sn \hookrightarrow A_{n+2}$ $\sigma \mapsto \sigma$ if $\sigma$ is even and $\sigma \mapsto \sigma (n+1, n+2)$ if $\sigma$ is odd.…
roi_saumon
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Permutations and Combinations Problem

I need help on the below problems. Thank you!! a. Intel will issue IDs for up to 5000 employees. ID Intel uses IDs beginning with a single capital letter followed by a string of numbers (if needed), what is the minimum number of digits necessary…
user59117
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How many times do all 3 clock hands align?

I have seen this question on several different websites and mathematics journals, but I haven't seen any concrete answers. For instance, some sources say it will happen only twice, while others say it will happen 22 times. Upon researching it, it…
Kenneth Sweezy
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Formula for permuted block sequences

In permuted block randomization, a block size of $4$ has $6$ possible permutations: $AABB, ABAB, ABBA, BBAA, BABA, and BAAB$ And a block size of 6 has 8 possible permutations: $AAABBB, BBBAAA, AABBAB, BBAABA, ABABAB, BABABA, ABAABB,$ and…
SEL
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The Least Natural number $n$ which has $18$ divisors

The Least Natural number $n$ which has $18$ divisors My Try:: $18 = 2\times 3 \times 3 = 3^2 \times 2$ Now How can I solve it Thanks
juantheron
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Multiplying permutation by itself prime times to get identity

Let $p \in\mathbb{N}$ be a prime number and $\alpha \in S_n$. If ${\alpha}^p=1$ then $\alpha$ has three disjoint options: $\ \bullet \alpha = 1$ $\ \bullet \alpha \text{ is a } p-cycle$ $\ \bullet \alpha \text{ is a product of } p-cycles$ Can…
Jose Fallas Rojas
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Counting of squares

A rectangle can be divided into $n$ equal squares. If the same rectangle can also be divided into $n+ 76$ equal squares then find $n$. I tried with taking dimensions of squares as $x$ and $y$. Then I made the respective lenghts and breadths equal…
maveric
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Compute the sign of a permutation

I am going through old exam questions for my upcoming exam, but got stuck on a question. Calculate $\text{sgn}(\tau)$ where $\tau = (4, 6, 7, 3, 5, 8, 1, 2)$. According to me, the sign of $\tau$ should equal $-1$, as the number of transpositions…
coder
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In how many ways can they be arranged such that any two green marbles won't be adjacent?

There are $2$ blue, $4$ yellow and $3$ green identical marbles. In how many ways can they be arranged such that any two green marbles won't be adjacent? _B_B_Y_Y_Y_Y_ Let us evaluate in how many ways blue and yellow marbles can be arranged. Since…
Mark
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