# Questions tagged [permutations]

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

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### Number of Permutations Question Given Values at their Specific Position

How many permutations of 6 digit string "abcdef" of the numbers 1 to 6 if I know a string has to be like "3bc5ef" with no repetition of the numbers? I know how to find the ...
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### Combination problem with exclusion [closed]

Two committees of 6 people is to be chosen from a group of 7 women and 5 men. How many committees are possible if those 5 mens must not be chosen on the same committee?
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### Everyone is passed everything exactly once, but never from the same person

Say 4 or more people are sitting around a table. Each has a sheet of paper. Devise an algorithm to pass these papers between these people that guarantees: Each person passes and signs every piece of ...
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### Number of ways places can be filled

We have N places and a container with M types of item which has to be placed in N places. But we also given an array A where i th position indicate that i th type of item cannot be placed A[i] times ...
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### prove following combinatorial identity for $(nPk)^2$

I have been trying to prove following thing. With no success prove that for $n>k\geq 2$ $$\sum_{i=0}^k nP_{(k+i)} . \binom{k}{k-i} kP_{(k-i)} = (nP{k})^2$$ where $nPk = \frac{n!}{(n-k)!}$ any ...
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### Work team sitting at a table question

A question came up recently: There is a team sitting at a round table. They all leave the table for a lunch break, and after returning to the table, each person finds that both their neighbours on ...
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### Permutation & Combination Problems [closed]

hope everyone are doing well. Just bumped into this question and I'm not sure on how to do it properly. Hope someone can give insight. Thank you so much. Question's Instruction Now, assume that there ...
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### Probability that you win in rock scissors paper with 3 players?

Let's say person A, person B and person C are playing rock paper scissors. Clearly $n(S)=3\times 3\times 3=27$. We want to find the probability that person A wins. $n(\text{only A wins})=3$ (A wins ...
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### Choosing with replacement with minimal allocation

If I have $r$ items and I am making $n$ distinct choices with replacement, then there are $r^{n}$ possible selections. However, how could I count with the added constraint that each of the $r$ items ...
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### Finding the cardinality of the centralizer in the permutation groups.

Let $\sigma\in S_n$ be a permutation. Find a formula (in terms of the factorization of $\sigma$ into disjoint cycles) for the cardinality of $C_{S_n}(\sigma)$. Fix $n$; for which permutations $\sigma$ ...
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### How Many 4-Digit Positive Even Numbers Use Only Digits 0 Through 4, With No Digits Repeating?

I am currently working on this problem: How many 4-digit multiples of 2 use only digits 0 through 4, with no digits repeating? I understand/know the basic idea of how to solve this kind of problem: ...
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### Permutations of race positions

Recently, I got this math question: Alice, Bob, Caroline, and Derek are competing on a 400 m run. a) How many different ways are there for them to arrive if participants can have joint places? b) Is ...
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### Why does the representation theory of $S_n$ fit perfectly with young tableaux?

This is a "soft question" of sorts. I've been studying the representation theory of the symmetric group, and I continue to be bewildered. Why does the representation theory of $S_n$ relate ...
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### Torsion group and its subgroup [closed]

Need the proof the following result. Q. If G is torsion group (i.e., every $g\in G$ is of finite order), then every subgroup of G is normal.
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### Asif wants to build tower of cubes

Statement: Asif has cubes of three colors. He builds a tower from them, placing each next cube on the previous one. It is forbidden to use more than $4$ cubes of each color. Asif finishes building ...
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### How can i calculate this combinations sum? [duplicate]

$C(n,2)+2×C(n,3)+3×C(n,4)+...+(n-1)×C(n,n)$ By calculating the combinations and multiplying , then simplifying i can until here: $(n-1)[n/2 + n(n-2)/3 + n(n-2)(n-3)/8 +...+ 1]$ I thought of ...
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### Leibniz determinant formula - Group Theoretic generalization.

This question rose from pure curiosity (in accordance with my current research). Given a matrix $A=(a_{i,j})$ of size $n\times n$ we have the butiful way to calculate its determinant by summing over ...
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### Alternative proof of Cayley's theorem works only for abelian groups

I am trying to prove Cayley's theorem in a way which avoids any particular maps from $G$ to a group of permutations on any particular set (contrary to the standard way of proving it, which involves ...
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### Rolling of dices [closed]

4 dices are rolled, find the number of outcomes in which ''exactly'' 2 dices show 3. I am emphasing the word "exactly". I can find the possibility in which at least one 3 will show up but ...
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### Number of ways to choose $4$ items from $6$ the same items such that order doesn't matter using the number of ways when the order does matter

I have 2 problems and I was able to solve each. Problem 1 How many ways you can draw 4 items from a box containing $6$ indistinguishable items such that the order is important? My answer is $6^4$. ...
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### Permutations of rolling of dices [closed]

4 die are rolled, find the number of outcomes in which ''exactly'' 2 die show 3. I can find the possibility in which at least one 3 will show up but couldn't find a way in which exactly two threes ...
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### How many possible ways are there to print 125 pages on 5 printers?

I think I understand this problem but I want to make sure I do get it thoroughly. In my mind, this problem is asking you to assign a single page to a single printer, but it doesn't really matter the ...
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### Reproducing methodology of probability model that calculates final race rankings

I am reading the paper of Graves, Rese and Fitzgerald. They propose a model for permutations which closely resemble the behavior formula 1 racing results. On page 5 they describe their method. ...
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### A true and false test has 10 questions. How many permutations of answers can be created, if at least 7 answers are true? [duplicate]

10 true or false questions. At least 7 answers are true. How many permutations can be created?
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### Unexpected result on the number of permutations with a restriction.

Let $p=(p_1,p_2,\dots,p_n)$ be a weak composition of a positive integer number $n$ into $n$ non-negative integer parts and let $k_i$ be the count of the part $i$ ($i=0,1,2,\dots$) in the composition. ...
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### Prove that $(B,S)$ is a BSGS, with $B = [1,2]$ and $S = [(1, 2, 3), (2, 3, 4)]$

Let $B = [1,2]$ and $S = [(1, 2, 3), (2, 3, 4)]$, I want to prove that $(B,S)$ is a Base and a Strong Generating System (BSGS). According to GAP, $(B,S)$ is indeed a BSGS. But I would like to prove ...
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### a false statement has 10 questions. how many permutations of answers can the teacher create if at least 7 answers are true [closed]

a true and false test has 10 questions. How many permutations of answers can the teacher create if at least 7 answers are true?
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### Given two identical groups with X distinct entities, and randomly selecting Y entities. What are the chances each selection contains at least 1 match?

My wife asked a version of this question because she was curious about the probability of getting matching villagers in animal crossing. I was curious to know the generalized method of solving this ...
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### How many cameras the police needs if there are $75$ crossroads in the town. [closed]

The traffic police is going to place telecameras on some crossroads, each camera controls the crossroad where it is located as well as the segments of intersecting streets including the neighbouring ...
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### Representation of the action of $S^3$ on $(1,2,-3) \in M$ On the plane

Let's consider the action of permutation of $S^3$ on the vector space $M =\lbrace (\lambda_1, \lambda_2, \lambda_3)\in \mathbb{R}^3, \lambda_1+\lambda_2+\lambda_3=0\rbrace$. The problem I have is the ...
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### Number of possible matrices using given conditions

I have a question, that goes as follows: The number of all matrices $A=[a_{ij}], 1 \leq i,j \leq 4$ such that $a_{ij}= \pm1$ and $\sum_{i=1}^4a_{ij}= \sum_{j=1}^4a_{ij}=0$ is? So, I think this ...
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### There are 100 towns in some country and each two towns are connected by a one-way road.

There are 100 towns in some country and each two towns are connected by a one-way road. we are to Prove that one can change the direction of at most one road so that after that each town will be ...
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### number of permutations in which all number changes

What is the number of permutations in which all number change position? For example, suppose a ordered set (12345678...n) What is the number of permutations this (below) is not allowed: Let P be, let'...
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### Counting permutations $\sigma \in S_n$ such that $\sigma$ is a product of $k$ disjoint cycles of length $r$.

Problem: If $kr \leq n$, with $1<r\leq n$, then the number of permutations that are products of $k$ disjoint cycles of length $r$ is $$\frac{1}{k!}\frac{1}{r^k}[ n(n-1)\cdots(n-(kr-1)]$$ I already ...
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### Prove that products of permutations $ab$ and $ba$ have the same cycle length
For example: Let $a,b \in S_3$. $a = (1,2,3), b = (1,2)(3)$, then $ab = (1,3)(2)$ and $ba = (2,3)(1)$. These are two different permutations, but their cycle type is the same. How could we prove this ...