# Tagged Questions

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

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### Question of P &C

There are 3 pots and 3 coins. All thesecoins are to be distributed into these pots where any pot can contain any number of coins. In how many ways all these coins can be distributed such that no pot ...
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### Tournament Of The Towns King and the 1000 wizard's

So i was doing one of the question's of TOURNAMENT OF THE TOWNS and I was not able to understand the solution given by them. The problem is: The King decided to reduce his Council consisting of ...
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### Team grouping troubles

Imagine there are 12 teams, numbered 1 through 12. There are 10 games those teams can compete in, with two teams needed per game. There are 10 rounds, and it is important that after the 10 rounds are ...
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### How many elements in $S_{8}$ are conjugate with $(12)(345)$?

How many elements in $S_{8}$ are conjugate with $(12)(345)$? My reasoning is as follows: Two elements in $S_n$ are conjugate if and only if they have the same cycle type, so we need to count the ...
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### Permutation in group theory

I am confuse how to proceed for the following question. Can you please help me. Thanks in advance! For a permutation $\pi$ of $\{1,\cdots,n\}$, one say that $k$ is a fixed point of $\pi$ if and only ...
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### Probability of increasing order permutation

Suppose I have n elements. What's the probability of a permutation such that the first half is increasing and second half can be ordered without any constraints? (A permutation can only have distinct ...
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### How to find the N-th 3 word sequence within the following constraints

I have a list of words. Let's say that I have an algorithm(explained below) to generate the permutations in a specific order. I want to be able to find the N-th permutation easily. I want to make ...
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### How to find the average Kendall's distance between 2 rankings

Suppose I have 2 rankings: $1$, $2$, $3$ and $2, 1, 3$ then the Kendall's distance between the two is 1 since there is only one pairwise adjacent switch. My question is, suppose my 2 rankings each ...
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### Why is some power of a permutation matrix always the identity?

If you take powers of a permutation, why is some $$P^k = I$$ Find a 5 by 5 permutation $$P$$ so that the smallest power to equal I is $$P^6 = I$$ (This is a challenge question, Combine a 2 ...
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### Finding the number of possible shortest ways. [on hold]

Find the number of possible shortest ways from A to B.
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### Number of positive unequal integer solutions of $x+y+z+w=20$

What is the number of positive different integer solutions of $x+y+z+w=20$, where $x,y,z,w$ are all different and positive? It would be nice if coding is not used. I am given the answer $552$.
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### Sylow subgroup of a symmetric group

consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutation.Let H be a 7-Syowl subgroup of$A_{20}$.Is that H must be cyclic? And is this statement correct ...
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### 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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### 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 ...
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### 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 ...
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### 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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### 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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### 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 ...
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### 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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### 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 ...
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### 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 $gHg^{-1}$. ...
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### 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 ...
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### 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 ...
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 $... 1answer 33 views ### Order of product of non-disjoint cycles Let$a$and$b$be two non-disjoint cycles of order$m$and$n$. Is there any general formula for the order of$a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ... 1answer 20 views ### N objects among K persons In how many ways can we distribute N objects among K people such that each person recieves AT LEAST ONE object ? Also the SUM MUST BE EQUAL TO N. eg. 7 objects can be distributed among 5 people in 2 ... 3answers 679 views ### Prove this using counting techniques:$\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 ... 2answers 24 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. ... 1answer 39 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 ... 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$(3^{4}-1)(3^{4}-3)(3^{4}-3^{2}).$... 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 ... 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 ... 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$(124) = (324)(132)(324)^{-1}$, ... 0answers 22 views ### number of inversions in permutation if subarray of permutation is reversed? I have permutation(P) of numbers 1 to N (<=10^5) . Suppose I can reverse the subarray of ... 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 ? 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 ... 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 ... 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 ... 3answers 45 views ### Suppose that an ice-cream café has 10 different flavors of ice cream. [closed] In how many different ways one can choose 3 scoops of ice-cream, so that order of flavors does not matter? 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 ... 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 ... 2answers 316 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 ... 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 ... 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 ... 4answers 333 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 ... 4answers 75 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 = ...
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 ...