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

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2answers
21 views

Number of permittable numbers given following conditions.

What are total numbers belonging to $\mathbb Q$ (rational) between $2008$ and $2009$ such that after decimal point their digits occur in decreasing order? \begin{align} 1) &\ 9Pi;i\in [1,9], \\ 2)...
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5answers
55 views

Arrange black and white balls so that each pair of white balls is separated by at least two black balls

I am trying to solve the following question: How many linear arrangements of $m$ white balls and $(n-m)$ black balls are possible such that each pair of white balls is separated by at least two ...
1
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1answer
23 views

Betting ended after nth round.Find the sum of money NOT WON?

Rahul and Vijay are playing a game with 12-sided die,where both of them lay bets on outcomes of roll of die.They start betting Rs 5 each on first round of the game and the amount bet in each ...
39
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5answers
2k views

Why is the determinant defined in terms of permutations?

Where does the definition of the determinant come from, and is the definition in terms of permutations the first and basic one? What is the deep reason for giving such a definition in terms of ...
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1answer
43 views

How to approximate the Langford numbers with probability?

A Langford pairing, also called a Langford sequence is a permutation of the multi set {$1,1,2,2, \dots, n,n$} in such a way that there are exactly $k$ elements in between every $k$. Interestingly, ...
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1answer
41 views

Monotone subsequence in a random permutation

I wish to compute the probability of having a log(n) length consecutive monotone subsequence in a random permutation of {1,...,n} (log with base 2). I'm trying to show it's $\leq1/n$, does it make ...
0
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1answer
60 views

Probability for a random permutation

Given a set of $N$ elements and a uniformly distributed random number generator, which always generates values between $0$ and $N-1$. Then the probability to get a random permutation (without re-draws)...
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1answer
19 views

Permutations (Making numbers from digits) [on hold]

Using the digits 1, 2, 4, 5, 7, and 8, how many different three-digit numbers can you form if each digit may be repeated any number of times in a number? I have tried to do this question and tried ...
2
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0answers
34 views

Orbits of the permutation action of a subgroup on its cosets

Consider a finite group $G$ and a subgroup $H \subseteq G$. There is a transitive group action of $G$ on the set of left cosets $gH$ by left multiplication, and the stabilizer of $gH$ is $gHg^{-1}$. ...
0
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1answer
29 views

How would I calculate the total number of combinations [on hold]

Lets say I have 4 lines or rows lets call them Row 1 .. Row 4 Now the total number of ways to delete the rows are: Row 1 (leaving Row2, Row3, Row4) Row 2 (leaving Row1, Row3, Row4) Row 3 Row 4 ...
3
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1answer
29 views

Generic method to distribute n distinct objects among r people such that each person gets at least one object

Is there any generic method to solve problems of the kind - "How many ways to distribute n distinct objects among r person(s) such that each person gets at least 1 object?". I am aware of 2 different ...
16
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2answers
595 views

Minimal generating set of Rubik's Cube group

The Rubik's Cube group is generated by the six moves $\{F,B,U,D,L,R\}$. However, is this the minimal generating set for the group? In other words, can I simulate the move $F$ just by making the moves $...
0
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1answer
33 views

Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
0
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1answer
20 views

N objects among K persons

In how many ways can we distribute N objects among K people such that each person recieves AT LEAST ONE object ? Also the SUM MUST BE EQUAL TO N. eg. 7 objects can be distributed among 5 people in 2 ...
6
votes
3answers
677 views

Prove this using counting techniques: $\sum_{k=0}^{n}{\binom{2n+1}k} = 2^{2n}$

I recently came across a question while studying for an exam. I haven't been able to solve it. We had to prove: $$\sum_{k=0}^{n}{2n+1\choose k} = 2^{2n}$$ We had to use counting techniques. This was ...
1
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2answers
21 views

Permutation of coefficient with coditions [on hold]

I have 6 coefficients, (V1,V2,H1,H2,D1,D2). Their permutation is 6! = 720. But I have a rule: V2 cannot lead V1, H2 cannot lead H1 and D2 cannot lead D1. For example: V2V1H1H2D1D2 is prohibit. ...
3
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1answer
38 views

Permutations of $S_7$

Find all permutations $\alpha \in S_7$ such that $\alpha^3 = (1 2 3 4)$. My attempt: We know that such an $\alpha$ must "look like" $(1432)$, since $(1432)^3=(1234)$. I think I need to find the ...
5
votes
1answer
29 views

Number of $4\times3$ matrices of rank 3 over a field with 3 elements.

I am finding number of $4\times3$ matrices of rank 3 over a field with 3 elements. If i count it as number of linearly independent columns i.e $3$ then its answer is $(3^{4}-1)(3^{4}-3)(3^{4}-3^{2}).$ ...
1
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1answer
63 views

Number of permutations with two elements in one cycle

Show, that for a set of permutations of a set $\{1,\dots,n\}$ $(n>0)$ the following statement is true. statement: The number of permutations where $1$ is in the same cycle with $k$, and the number ...
3
votes
1answer
28 views

Bell numbers and the Moments of expected number of fixed points

Let $X_N$ be the random variable corresponding to the number of fixed points (1-cycles) in a permutation chosen uniformly at random from $S_N$. Then, the $m^{\text{th}}$ moment, when $m < N$, is ...
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3answers
30 views

Finding all normal subgroups of $A_4$

I was reading up on this: Find the number of normal subgroups of $A_4$. If $H$ has a $3$-cycle, say $(123)$, then $H$ has its inverse $(132)$ thefore it also has $(124) = (324)(132)(324)^{-1}$, ...
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0answers
21 views
1
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1answer
26 views

20 identical balls to be distributed in 3 identical boxes with MAX & MIN balls in each box?

As the title suggests, In how many ways can 20 identical balls be distributed in 3 identical boxes with at most 8 balls in each box and minimum 1 ball in each box ?
1
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2answers
51 views

Scheduling a Round Robin tournament - 4-way games

I'm looking to schedule 16 players to play a round robin tournament with each other such that there are 4 players at each table. I'd like for each player to play with each other player exactly once ...
1
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1answer
46 views

Notation for probability: $C_n^r$, $P_n^r$, $A_n^r$?

I was told that $C^{n}_{k}$ refers to combinations or choose k elements from n elements, $\bar{C^{n}_{k}}$ refers to combinations with repetitions (i.e. $C^{n+k-1}_{k}$), and $P^{n}_{k}$ refers to ...
0
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4answers
37 views

Why doesn't this alternative method work? Chance of getting four of a kind in a hand of $5$ cards?

Please note: This is not a duplicate since it is asking about an alternative method of solving the question What is the probability of getting four of a kind in a hand of $5$ cards from a standard ...
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votes
3answers
44 views

Suppose that an ice-cream café has 10 different flavors of ice cream. [on hold]

In how many different ways one can choose 3 scoops of ice-cream, so that order of flavors does not matter?
6
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3answers
58 views

In how many ways can an inspector visit $4$ normal sites and $1$ “suspicious” one?

I cannot figure out why my answer to the following question is wrong: Suppose that a weapons inspector must inspect each of five different sites twice, visiting one site per day. The inspector is ...
1
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0answers
65 views

How many possible solutions are there in 4 number game?

$4$ number game consist of $4$ random number from $0$ to $9$. The goal is to make the result equal to $24$. There are only operation four operation possible, addition, subtraction, multiplication, and ...
0
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2answers
313 views

When a 0-1-matrix with exactly two 1’s on each column and on each row is non-degenerated? [1]

Let $A$ be an $n\times n$ matrix with entries in the set $\{0,1\}$ which has exactly two ones in each column and two ones in each row. Give necessary and sufficient conditions for the rank of $A$ to ...
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2answers
2k views

Find the center of the symmetry group $S_n$.

Find the center of the symmetry group $S_n$. Attempt: By definition, the center is $Z(S_n) = \{ a \in S_n : ag = ga \forall\ g \in S_n\}$. Then we know the identity $e$ is in $S_n$ since there is ...
0
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1answer
65 views

How do I calculate all possible combinations for a player creator in a game?

I'm currently working on a character creator for a game, but I don't know how to calculate all possible character combinations the player can create. In the creator, the player is required to choose ...
2
votes
4answers
332 views

Distributing Objects into Boxes (Discrete Mathematics)

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 ...
0
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4answers
73 views

How many $3$ member subsets $\{x, y, z\}$ of positive natural numbers have the sum $x + y + z = 100$?

I have a math homework problem where I think I have to use Permutation/Combination. The question is: How many $3$ member subsets $\{x, y, z\}$ of positive natural numbers have the sum $x + y + z = ...
2
votes
4answers
93 views

Number of positive unordered integral solutions

What are the number of positive unordered integral solutions for $a+b+c=36$ Solution given is $108.$.But I am getting $91$ as $$\frac{\binom{35}2-3\times16-1}{3!}.$$ $3\times16($ for $a=b$ cases and ...
0
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1answer
21 views

Permutation/Combination Assistance

I'm stuck on the following permutation/combination problem... John won 10 tickets to the Falcons/Ravens game by calling into a radio station’s contest. Find the number of ways he can invite family ...
7
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1answer
632 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: ...
3
votes
1answer
56 views

n points permuted on a circle

Here is a combinatorics problem that bothers me a lot. I am looking forward to a quick reply. Thanks in advance. Here goes the problem. Initially there are $n$ points on a circle. We do permutation to ...
0
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1answer
17 views

Proof that a transitive permutation group (G, X) with G abelian, is sharply regular

As the title states, the question is the following: Let (G, X) be a transitive permutation group, where G is abelian. Show that (G, X) is "sharply regular". First of all I want to notice that in my ...
2
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4answers
53 views

Circular Arrangement with numbers

The number of ways of arranging 2 women and 7 men around a circular table containing nine numbered chairs such that the women are not together. I am getting answer as 7!*7c2(arranging 2 women in the ...
2
votes
1answer
27 views

Limit of probability that a permutation of $\mathcal{S}_n$ has a $k$-cycle is $1 - e^{-1/k}$?

Choose a random permutation $\sigma \in \mathcal{S}_n$. What's the probability that it contains a $k$-cycle as you take $n \to \infty$? I ran a couple examples and it seems to approach $1 - e^{-1/k}$....
1
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1answer
52 views

Dice rolls - Combinatorics with limitations

Given 2 players, one rolling $x$ d6 dice and the other rolling $y$ d6 dice, what is the probability of a match between the two players? I'm getting stuck on the sub-set comparisons - I can calculate ...
1
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1answer
30 views

Trying to find the rank of the word permutation .

What is the rank of the word $PERMUTATION$ if all the words formed by the letters of "$PERMUTATION$" are arranged in ascending order ? $PERMUTATION$ in ascending order in $\{A,E,I,M,N,O,P,R,T,T,U\}$ ...
3
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4answers
2k views

Number of ways to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball

How many ways are there to distribute 5 distinguishable balls between 3 kids such that each of them gets at least one ball? My approach is $ \binom{5}{3} 3! $ + $ \binom{2}{2} \binom{3}{2}2!$ which ...
25
votes
6answers
39k views

Combination of smartphones' pattern password

Have you ever seen this interface? Nowadays, it is used for locking smartphones. If you haven't, here is a short video on it. The rules for creating a pattern is as follows. We must use ...
20
votes
5answers
359 views

Show that : $\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2= 2 n!$

I came across this result while doing some representation theory of the permutation group $S_n$ $$ \sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
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0answers
23 views

Is this statement permutation or combination? [closed]

How many ways can a person select five different numbers between 1 and 14 (inclusive)? (1 mark)
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1answer
505 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? ...