# Tagged Questions

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

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### How to calculate a pair of cards contains at least one ace?

A pair of cards are simultaneously drawn from a deck of 52 cards three times in a row. The drawn cards are returned to the deck. What is the probability that two of three pairs contain an ace? For ...
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### How many ways there are to arrange 40 people to play exactly one match each? [duplicate]

A tennis club has 40 members. They host a tournament playing single (one verses one) matches. Every member of the club plays one match with another member of he club, so twenty matches are ...
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### Are there any Symmetric Groups that are cyclic?

Are there any Symmetric Groups that are cyclic? Because I have been doing some problems and I tend to notice that the problems I do that involve the symmetric group are not cyclic meaning they do not ...
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### Nearest latin square

given a n x n matrix A with integer entries is there any way to find the nearest n x n latin square to it, say, e.g., in the Frobenius norm? I am looking for some type of convex optimization... ...
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### Permutations to satisfy a challenging restriction

In a stack of n distinct cards in order {1,2,3,4,...,n} from top, define distance between 2 cards as the number of cards between them. 2 cards are neighbours if they're adjacent in original ...
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### Find a subgroup of $S_{4}$ which is isomorphic to $\mathrm{Aut}(U_{8})$

The notation I am using is: $S_{4}$: the permutation group of order 4 $\mathrm{Aut}(U_{8})$: the set of all automorphisms on the set $U_{8}$ $U_{8}$: the group of numbers relatively prime to 8 I ...
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### How many permutations of a multiset have a run of length k?

Background $\newcommand\ms[1]{\mathsf #1}\def\msP{\ms P}\def\msS{\ms S}\def\mfS{\mathfrak S}$Suppose I have $n$ marbles of $c$ colors, where $c≤n$. Let $n_i$ denote the number of marbles of color ...
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### Arranging pictures possible combinations

I'm working on a problem which states there are 26 portraits of men and 4 of women. It wants to know how many ways can the photos be organized so no women are next to each other. I assume that the ...
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### Dividing gems by random permutation

A group of people have found a treasure of gems: $G=90$ green and $B=990000$ blue. They decided to divide it among them. Since there are more people then gems, they decide to order themselves in a ...
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### Group Theory - Permutations

If $B \in S_7$ and $|B^3| = 7$, prove that $|B|=7$. Solution: As $o(B^k) = o(B) / (o(B),k)$ Thus $|B| / (|B|,3) = 7$ Let $|B| = 7a$. Then $7a/(7a,3)$ should be $7a/a = 7$ or $(7a,3) = a$. As $3$ is ...
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### Proof of 2^n deck of card, it will be reverse order performing n perfect in-shuffle.

I am now trying to prove performing n perfect in-shuffle with 2^n deck of card, and then it will be resulting reverse order. For example, Initial : [1, 2, 3, 4] 1st round : [3, 1, 4, 2] 2nd round : ...
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### how many strings you can write with the letters abcd (permutation & combination or what?)

You have 4 letters abcd. How many 4-letter strings can you write with them? Assumptions: - the order is not important (aaab, abaa, baaa are same, counts 1) - you can use same letter more than once ...
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### Permutations on a set with certain conditions.

Suppose we have a set $S=\{1,2,3,x,y\}$. There are $5!$ ways to rearrange the elements in the set, but I am confused about how to find the number of ways to rearrange the set given that $3$ comes ...
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### How to prove this equality for Stirling numbers?

How can I prove that the following formula is true for Stirling numbers of first kind. $$\sum_{k=1}^n(-1)^k\left[\begin{matrix} n\\k\end{matrix}\right] =0$$ Actually I want to prove that number of ...
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### Counting with permutations and counting ignoring permutations

I am given this problem: This problem was given to me in my computer science class but it has to do with permutation and I want to understand it mathematically first. let $c(n)$ be the number of ...
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### Four Letter-envelop problem

A secretary writes four letters and the corresponding addresses on envelopes. If he inserts the letters in the envelopes at random irrespective of the addresses, (i) find the probability that only one ...
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### Number of visible elements in a permutation

The following problem occurred to me the other day, and I've played around with a bit but can't seem to find a good solution: Consider a permutation $\pi$ of $\{1, 2,\ldots ,n\}$. For every positive ...
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### probability of vector in column span

Consider we have a fixed matrix M of size a$\times$ 2b (Let us look at M=[$M_1$ $M_2$] where matrices $M_1$,$M_2$ are of size a$\times$b) and a vector $v$ of dimension a. Is there any way that I can ...
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### Outer automorphisms of the infinite symmetric group

Denote by S$_\infty$ the group of permutations of $\mathbb N$. Question: Does there exist an outer automorphism of S$_\infty$, and if so, can one be exhibited? Does this depend on the continuum ...
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### Generators of symmetric group $S_n$ [duplicate]

How can you prove that $S_n$ is generated by $(1\space 2)$ and $(1\space 2\space 3\space ... \space n))$ for $n\geq 2$?
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### Finding cycles with set of permutations

Let $\alpha = (\alpha_1 \, \alpha_2 \, \ldots \, \alpha_s)$ be a cycle, for positive integers $\alpha_1 , \alpha_2 , \ldots , \alpha_s$. Let $\pi$ be any permutation. Show that $\pi \alpha \pi^{-1}$ ...
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### Does the product of all conjugates of some subgroup is independent of the order?

Let $G$ be a finite group and $A \le G$. Let $A^G = \{ A_1, A_2, \ldots, A_n \}$ be all the conjugates of $A$, i.e. each $A_i$ equals $A^g$ for some $g \in G$. Then I want to show that  A_1 A_2 ...
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### permutations of 10 objects in a subset contains similar elements

A board that is divided into 15 different places, and we want to place 10 components on this board such that each component is placed in a section; knowing that those components are divided into 4 ...
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### Simple Probability Question about Combinations

If someone could please point me in the right direction on these. I get lost on how to think about them. In a game there are four holes with values 0, 1, 2, and 4. You are given 6 balls to shoot into ...
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### Double Check Probability for Permutation

I have to find the sample space and a few probabilities here and I am wondering about if I am going down the right track for these. If I am incorrect, then please point me in the right direction, but ...
### Question about permutations: How to show $\sigma(P)=(-1)^{\imath(\sigma)}P$?
A permutation of a finite set $X$ is any bijection from $X$ to $X$. We denote by $S(X)$ the set of all permutations of $X$. If $I_n:=\{1, \ldots, n\}$ we write $S_n$ instead of $S(I_n)$. Define ...
I know the definition of parity of permutation. But what does that look like in examples? For example, if the number of permutations is odd, then the sign of permutation in $-1$. What does this mean? ...