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

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1answer
19 views

Distribute 20 million $ among 4 companies with some constraints

20 million is to be invested in 4 companies A, B, C, D. The minimum amount for investments are 1, 2, 3, 4 million respectively. How many different investment strategies are available if An ...
3
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1answer
33 views

In how many ways we can arrange 12 people in a row if 5 men are constrained to sit next to each other together?

In how many ways we can arrange $12$ people in a row if $5$ are men and they must sit next to each other? My approach I consider $5$ men as one entity and so now there are $8$ people to be seated ...
0
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0answers
20 views

A kind of permutations and possible relation to cyclic groups.

Any permutation that moves $n$ elements in some fashion never revisiting the same until all others have been visited, in other words so that: $${\bf P}^n = {\bf I}, \text{ but no } 0<m<n \text{ ...
2
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5answers
710 views

How many possible “words” can be made from the first seven letters of the alphabet, allowing for repetition and enforcing alphabetical order?

Using letters from the alphabet $A = \{a, b, c, d, e, f, g\}$, how many words of length $5$ are possible when repetition is allowed but the letters must occur in alphabetical order? Not sure how to ...
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3answers
27 views

Find the number of ways to arrange 8 students with restriction [on hold]

8 students are arranged in a row. How many ways to arrange them if 3 particular students must be separated?
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0answers
14 views

Discrete subgroups of $SU(m) \times SU(n) \times U(1)$

Is anything known about which permutation groups are subgroups of $SU(m) \times SU(n) \times U(1)$? Alternatively, how can one possibly go about finding them. I am trying to find discrete groups of ...
0
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1answer
34 views

Permutations of geometric structure

Sorry about the title, i don't know how to describe this problem. I tried counting my way through this problem but kept getting the wrong answer(which is 12, by the way). Is there a more systematic ...
2
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0answers
29 views

Is there a method that determines an unknown permutation better than $\sum_{k=1}^n (k+1)/2$ steps on average?

Suppose I have a random permutation $s \in S_n$ that is unknown to me. However, suppose I can make a query where when I ask if $i$ is in the $j$th position in the permutation, I receive a yes or no ...
2
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1answer
89 views

Finding permutation with a condition

Let $a$ be a permutation in $S_6$. I'm asked whether there is an $a$ so that $a^2 = (123)(456)$ I'm quite confused about where to start. I do know the $a$ must consist of $3$ elements (right?). How ...
7
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1answer
639 views

$N^\text{th}$ (in lexicographical order) term of balanced brackets string

We have the following balanced brackets permutations of length $4\cdot2$ in lexicographical order: ...
2
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2answers
57 views

Normal subgroups of $A_5$ must contain a 3-cycle.

I am trying to prove the simplicity of $A_5$ by showing that every non-trivial normal subgroup $H$ contains a 3-cycle, and therefore is all of $A_5$ since the 3-cycles all belong to one conjugacy ...
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0answers
38 views

what are the number of ways to select a 4 digit number with a 3 digit number always included? [on hold]

Number of ways to select 4 digit number( X X X X ) should have three digit number ( say 1 2 3 ) It should be in same order.
0
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1answer
69 views

Number of sequences that maintain a property

In how many ways can i create a sequence of $m$ elements from the set $1,2,...,n$ such that the longest strictly increasing subsequence of it is exactly $n$? For example if $n=3$ and $m=4$ then the ...
0
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2answers
23 views

Formation of Teams in Permutation and Combination

A class has $n$ students , we have to form a team of the students including at least two and also excluding at least two students. The number of ways of forming the team is My Approach : To include ...
0
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1answer
42 views

In how many ways can $5$ Indians, $4$ Chinese, and $3$ Americans be assigned to $12$ stations so that no two Americans serve at consecutive stations?

On a railway route from Delhi to Jaipur there are $12$ stations . A booking clerk is to be deputed for each of these stations out of $12$ candidates of whom $5$ are Indians , $4$ are Chinese and the ...
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4answers
46 views

How many mixed double pairs can be made from 7 married couples provided that no husband and wife plays in a same set?

So for first man there can be 7 possible partners including his wife, for the next man there will be 6 possible partners and so on, therefore for $7$ men and $7$ women, there will be $7!$ possible ...
3
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1answer
17k views

possible combinations of 3-digit

How many possible combinations can a 3-digit safe code have? Because there are 10 digits and we have to choice 3 digits from this, then we may get $10^P3$ but A author used the formula $n^r$, why is ...
1
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1answer
48 views

Elements that are their own inverses in a symmetric group.

How many elements are their own inverses in $S_6$? I'm having a hard time figuring out how to calculate such a thing.
0
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1answer
33 views

counting number of steps using permutation-combination

We need to climb 10 stairs. At each support, we can walk one stair or you can jump two stairs. In what number alternative ways we'll climb ten stairs? How to solve this problem easily using less ...
1
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3answers
41 views

Identical Obects in Permutation and combination

There are $2$ identical white balls , $3$ identical red balls and $4$ green balls of different shades. The number of ways in which the balls can be arranged in a row so that at least one ball is ...
0
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1answer
40 views
+250

Upper-bounding a sum over non-identity permutations

EDIT: Question 1 has been settled (below). The bounty is for question 2. Let $n\geq 3$ and consider the following function $f:S_n\backslash\{e\}\rightarrow \mathbb{R}$ $$f(\sigma)=\sum_{i=1}^n\frac{...
1
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1answer
24 views

Prove all products of two disjoint 2-cycles are pairwise conjugate in $A_{n}$

Let $n \geq 4$. Prove that in the group $A_{n}$ of parity-even permutations of $\{1,2,...,n\}$, all products of two disjoint 2-cycles are pairwise conjugate. This is a past exam question for a ...
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0answers
29 views

All 3-cycles are pairwise conjugate.

In my lectures on Group Theory our lecturer claimed the following: All 3-cycles are pairwise conjugate. He then went on to prove this but I am struggling with understanding his proof. I will try and ...
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4answers
2k views

How many numbers lying between 10 and 100 can be formed with the help of digits 1,3,5,7,9?

Please help me!! i don't know how resolve this problem. This is exercise from permutations in my school. hope you guys can help me :)
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2answers
30 views

Distinct digits in a combination of 6 digits

How many 6-digit numbers contain exactly 4 different digits? My approach is: For any 3 digis same and the remaining 3 different(aaabcd) 4*9*8*7*6 For any 2 duplicate digits(aabb) and the remaining ...
1
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2answers
64 views

How many 6-digit numbers contain exactly 4 different digits? [duplicate]

my solution is----> NUMBER can be 777210 this i denote by aaabcd ------ this can be done in ---> 10*1*1*9*8*7*[6!/3!] {1 for a thrice} NUMBER can be 772210 this i ...
0
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0answers
14 views

Writing in disjoint cycles

Write $\left ( 123 \right )^{-1}\left ( 23 \right )\left ( 123 \right )$ in disjoint cycles. I get that $\left ( 123 \right )^{-1}$=$\left ( 132 \right )$ $\left ( 132 \right )\left ( 23 \right )\...
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6answers
34 views

Arrangement of 12 boys and 2 girls in a row.

12 boys and 2 girls in a row are to be seated in such a way that at least 3 boys are present between the 2 girls. My result: Total number of arrangements = 14! P1 = number of ways girls can sit ...
2
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1answer
19 views

No subgroup of $S_n$ containing stabilizier of 1?

Is it true that the stabilizer of $1\in \left\{1,\dots ,n \right\}$ in $S_n$ is a maximal subgroup? Intuitively I'm thinking that as soon as you add another permutation, you'll somehow be able to ...
2
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0answers
28 views

Number of zigzag permutations of first $n$ natural numbers given start and end value

Given $n$ and $1\le s,e\le n$, how to compute the number of zigzag permutations of first $n$ positive integers starting with $s$ and ending with $e$? I tried formulating a recurrence relation but can'...
8
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0answers
95 views

How many ways can I arrange the numbers $1$ to $N$ with this divisibility condition?

For the numbers $1, \ldots, N$, how many ways can I arrange them such that either: The number at $i$ is evenly divisible by $i$, or $i$ is evenly divisible by the number at $i$. Example: for N = 2$...
0
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2answers
27 views

Permutations & series [closed]

Consider all the $7$ -digit numbers containing each of the digits $1,2,3,4,5,6,7$ exactly once, and not divisible by $5$. Arrange them in decreasing order. What is the $2015$th number (from the ...
2
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1answer
22 views

Concept of alike in Permutation and Combination

Number of ways in which $7$ green bottles and $8$ blue bottles can be arranged in a row if exactly $1$ pair of green bottles is side by side . (Assume all bottles to be alike except for the colour). ...
2
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2answers
48 views

Using the Binomial Identity, prove that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Using the Binomial Identity, prove that: $${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$$Because this is in the form of a Binomial Coefficient, I can break down the LHS further:$$\left(...
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2answers
47 views

How many integers can be formed by using exactly $x$ $4$'s, $y$ $5$'s, and $z$ $6$'s if no other numbers are used?

Can anyone tell me the total number of integers than can be formed by using exactly $x$ $4$'s , $y$ $5$'s and $z$ $6$'s and no other numbers are used? For $x=1$, $y=1$, $z=1$, the total is $6 \...
0
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1answer
27 views

Cyclic permutation group example ($n>1$)

I have googled around and haven't been able to find any examples of some $S_n$ with $n>1$ that is a cyclic group. This may mean it is a dumb question, any help is appreciated.
1
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1answer
521 views

In S4, find all the even permutation and show that the set of odd permutations isn't stable for binary operations in S4.

I want to find the even permutations of $S_4$ so i am supposed to find the transpositions right? but of what permutation exactly do i find the transpositions? And how do i know which ones are even? ...
3
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1answer
32 views

How to find number of integral solutions, containing large number of cases?

Number of positive unequal integral solutions of the equation $x+y+z=12$ can be found out knowing the cases it involves: $(1, 2, 9) , (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5)$. Thus, ...
4
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2answers
35 views

Permutation of 2 or more groups while keeping the ordering of the groups

I've been trying to get a general formula for this, but I couldn't find anything exactly what I need. What I want is, let's say we have 3 groups: (x,y,z),(a,b,c) and (k,l,m) What is the total number ...
2
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1answer
35 views

Can someone explain this proof of the relationship between chromatic number and independence number to me?

I came across the following claim and proof in this paper, and I really don't follow. If $G$ is a vertex-transitive graph with independence number $\alpha$ and chromatic number $\chi$ then $n/α(G) ≤...
2
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1answer
57 views

Condition for a binary matrix to contain a permutation matrix

I would like to know if there is any condition to check whether a binary matrix contains a permutation matrix of the same size. E.g. $$A_1=\pmatrix{1&1&1&1\\ 1&0&0&1\\ 1&0&...
0
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1answer
44 views

Combination Problem : $6$ Countries , $4$ players from each country

$6$ Countries participate a world tournament . Each country has $4$ players. One Cricket player , One Rugby player , one Volleyball player and one Football player. Need to select a team of $8$ ...
2
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1answer
50 views

Understand a part of the proof about permutations in a symmetric group on $n$ elements

Let $\sigma$ be an even permutation in $S_n$($\sigma \in A_n$). Assume $\sigma = \tau\sigma\tau^{-1}$ for some $\tau \in S_n$ and assume that the type of $\sigma$ consists of distinct odd integers. ...
0
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1answer
901 views

Circular Permutations With Repetitions (Mirrored Ignored)

For Circular Permutations with unique elements (mirrored ignored) the answer is (n - 1)!/2 (pretty straight forward). However I cant seem to figure out how to calculate circular permutations with ...
0
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0answers
22 views

Consider the system S which can take n input parameters and each parameter can take on m values

(a) What is the maximum number of pairs a single test case for this system can cover? "I know that there are m^n different combinations in this example, but i'm unsure how many pairs a single test ...
0
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0answers
23 views

Summation containing permutations.

Given $$a_1< a_2<......< a_n$$ find a permutation $\sigma$ maximizing the sum $$\sum_{i=1}^n {a_i \over \sigma(i)}$$ I can't figure our where to begin. I know that the solution is $\sigma=...
9
votes
2answers
276 views

Counting: how many ways of climbing a stair?

You are climbing a staircase. At each step, you can either make $1$ step climb, or make $2$ steps climb. Say a staircase of height of $3$. You can climb in $3$ ways $(1-1-1,\ 1-2,\ 2-1)$. Say a ...
0
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1answer
28 views

Permutation Product

Here is a problem from Algebra by Michael Artin: $p = \left( {\begin{array}{*{20}{c}} 3&4&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&5 \end{array}} \right),q = \left( {\...
0
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1answer
467 views

About permutation with repeated identical elements.

First up, I do know the general solution but somehow am unable to use it to solve this kind of problem. I am simply lost. The problem is like this: ...
2
votes
5answers
354 views

Packing $8$ identical DVDs into $5$ indistinguishable boxes

I am trying to solve this question: How many ways are there to pack eight identical DVDs into five indistinguishable boxes so that each box contains at least one DVD? I am very lost at trying to ...