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

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21 views

If there are 10 parents and 8 teachers nominated for positions on the school council, how many different committees can there be?

A school council consists of 12 members, 6 of whom are parents, 2 are students, the Principal and the remainder are teachers. The school captain and vice-captain must be on the council. If there are ...
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0answers
23 views

What is the proof of the formula for generalized permutations (permutations with finite repetition allowed)?

I have currently been studying discrete mathematics and combinatorics where I came across the introduction to generalized permutations in the textbook (Introductory Discrete Mathematics by V.K. ...
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1answer
40 views

The product of $(a_1-1)(a_2-2)…(a_{169}- 169)$ is [closed]

Let $a_1,a_2,.....a_{169} $ represent any arbitrary permutation of the number $1,2,3....169$. Then the product $(a_1-1)(a_2-2)......(a_{169} - 169)$ is Odd only for some permutation, not all Always ...
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0answers
28 views

Show $\sum_{k=0}^n b_r(n,k) = (r-1)!\frac{x^{\bar{n}}}{(x+1)^{\bar{r-1}}}$ [duplicate]

Let's define $b_r(n,k)$ as $n$-permutations with $k$ cycles where numbers $1\dots r$ belong to one cycle. I tried to first define closed form for $b_r(n,k)$. My idea: We need to put $1 \dots r$ into ...
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0answers
3 views

Boolean equivalence

Given a boolean formula $\phi$ and an interpretation $x$ that satisfies it, is it possible to come up with a permutation $P$ such that $x$ satisfies $P(\phi)$ but it is computationally hard to ...
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0answers
45 views

Combinatorial analysis

There are $20$ children in a lost ship. They do not remember their birthdays but would like to be assigned with one. 1. In how many ways this can be done so that exactly $2$ children will get ...
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1answer
50 views

Is this equation true?

As the question states, does this equation hold true? $\sum_{j=0}^n \sum_{E \in {n \choose j}} (-1)^{|E|}(n-|E|)! = \sum_{j=0}^n(-1)^j(n-j)!{n \choose j} $ From what I understand, this holds true at ...
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0answers
15 views

Solve $\max_X \mathrm{sum}(AXB \geq \gamma)$, with $X$ being a permutation matrix

I have a problem to find the best permutation matrix $X \in \{0,1\}^{n \times n}$, which would maximizes the number of elements in $AXB$ which are above a certain positive number $\gamma$. In other ...
11
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1answer
588 views

What is the rank of the matrix consisting of all permutations of one vector? [duplicate]

Let $a=(a_1,...,a_n)^\top\in\mathbb{R}^n$ be a column vector and let $M_1,...,M_{n!}$ denote all $n\times n$ permutation matrices. When is the rank of the matrix that consists of all possible ...
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0answers
22 views

How many combinations to break a monoalphabetic substitution

Let a language $\Sigma$ have 16 letters, we have a message in that language that was encrypted using monoalphabetic substitution (a permutation of the alphabet) and we want to break it. We also ...
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0answers
104 views

Number of ways to arrange $n$ numbers based on their relative values to each other

EDIT I've found a formula to solve this question, but I don't understand the reasoning behind it. Can someone explain this formula? $s(n - 1, x + y - 2) \times C(x + y - 2, x - 1)$ $s$ being ...
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0answers
18 views

Find permutation matrix $X \in \{0,1\}^{N \times N}$ in order to make $XAX \geq_c B$

I need to solve a problem to find out the best permutation matrix $X \in \{0,1\}^{N \times N}$ which would maximize the number of elements in matrix $XAX$ which are above (component-wise) matrix $B$ ...
3
votes
1answer
36 views

What is the probability that all books of the same language land next to each other in a random arrangement?

4 different Mathematics books, 3 different German books, and 3 different Spanish books are arranged randomly on a shelf. What is the probability that all books of the same language will land next to ...
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2answers
35 views

If five letters from the word SPECIAL are arranged randomly with no repetitions, determine the probability that the word SPICE will be chosen.

Given the word SPECIAL, determine the probability that the word SPICE will be chosen if the letters from "SPECIAL" are arranged randomly without repetitions. Thanks, in advance.
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0answers
25 views

enumerating all permutations of partitions

Given a set of $n$ items, I would like to enumerate all the permutations of all the partitions of those $n$ items. For example, in the case of $n=3$ (with Bell number 5), there are 13 permutations of ...
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3answers
37 views

How many fix point free permutations of 5 elements are there? [duplicate]

I am trying to find out how many fix point free permutations of 5 elements there are. A permutation is fix point free, if $\pi (i) \neq i$. I am trying to solve this problem using the inclusion ...
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1answer
29 views

Finding number of ways of selecting 6 gloves each of different colour from 9 pair of gloves? [duplicate]

There are nine pairs of gloves each of different colors in how many ways can we arrange six gloves such that each is of different color? I tried like this : First number of ways in which we can select ...
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1answer
35 views

Finding number of ways of selecting 6 gloves each of different colour from 18 gloves? [closed]

There are nine pairs of gloves each of different colors in how many ways can we select six gloves such that each is of different color? I tried like this : First number of ways in which we can select ...
-1
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1answer
29 views

Monge's shuffle questions on permutations

I am naive in number theory and having problems in finding the following exercise questions. Please some one help me with these questions. I want to thank you all in advance. :-) The first part of ...
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3answers
43 views

Distribution of $28$ objects into 4 heaps

In how many ways can $28$ different things can be formed into $4$ heaps so that each may contain $7$ things? Could someone give me slight hint for this particular question.
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1answer
70 views

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common?

Two squares are chosen at random on a chessboard. What is the probability that they have a side in common? I have got the total no of events by using 64 C 2. But I am unable to find the numerator(no. ...
1
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1answer
20 views

Listing all possible trebles from multiple sets where you can only select one element of each set

I'm stuck on a combinatorics problem and was hoping someone could help me. I would like to know the number of possible trebles (order not important) from sets of elements where only one element can ...
2
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4answers
83 views

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue?

A bag contains 2 red, 3 green and 2 blue balls. Two balls are drawn at random. What is the probability that none of the balls drawn is blue? I am helpless regarding this. I don't know how to solve it....
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0answers
19 views

how many different ways a set of ten variables could be ranked from highest to lowest value, given three possible integer values for each variable

I'm trying to solve a problem involving how many different ways a set of ten variables could be ranked from highest to lowest value, given three possible integer values for each variable. It's been ...
3
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0answers
81 views

Optimal cyclic permutations (Formulate as standard problem)

How can we find cyclic permutations $\prod_i$ to be applied to each of corresponding $i$'th rows of a square matrix $X$ of size $n \times n$ such that a given sum of pairwise costs $\sum_{ij}C\left[\...
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0answers
30 views

Number of distinct integer-valued vector solution for $x_1 + x_2 + … + x_r = n$ [duplicate]

The Number Of Integer Solutions Of Equations $$x_1 + x_2 + ... + x_r = n$$ An approach is to find the number of distinct non-negative integer-valued vectors $(x_1,x_2,...,x_r)$ such that $$x_1 + x_2 +...
3
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3answers
379 views

Arranging numbers around a square

In how many ways numbers 1 to 12 can be arranged on a sides of squares (5 places on each sides i.e 20 places total) leaving 8 places empty? I am getting answer as 12c5(selecting 5 numbers)*7c5(...
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2answers
68 views

Find the divisors of $5040$ in the Plato's dialogue “Theaetetus”

In the Plato's dialogue "Theaetetus", at a certain point, we have the following "problem" \begin{align*} 5040 &= 7! \\ &= 1\times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \\ &= 2 \...
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0answers
27 views

Shuffles vs direct sums of permutations

A $(p,q)$-shuffle is a permutation of $p+q$ things that preserves the internal order of the first $p$ things and of the last $q$ things. As remarked on wikipedia, since a $(p,q)$-shuffle is uniquely ...
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1answer
60 views

Another form of Menage Problem : Place 8 more cherries(maroon) removing berries(black) 1 from each row and each column. No of ways?

I tried to see it as a matrix where for a position (i,j) , i+j = 8, 9, 16 means you can't change that position. Any help?
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1answer
29 views

Question of P &C [closed]

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 ...
0
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1answer
26 views

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 ...
3
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1answer
68 views

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 ...
4
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2answers
53 views

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 ...
2
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3answers
22 views

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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4answers
98 views

Permutation in group theory [closed]

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 ...
2
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2answers
55 views

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 ...
3
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2answers
31 views

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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1answer
36 views

Finding the number of possible shortest ways. [closed]

Find the number of possible shortest ways from A to B.
0
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1answer
36 views

Sylow subgroup of a symmetric group

Consider the symmetric group of$S_{20}$ and it's subgroup $A_{20}$ consisting of all even permutations. Let $H$ be a $7$-Sylow subgroup of$A_{20}$. Is $H$ cyclic? And is correct the statement which ...
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2answers
819 views

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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2answers
61 views
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3answers
48 views

How to express/write a permutation of a Set?

How to express a permutation (without repetition) of a Set $A$? I'd like to create a set $P$ of tuples while equal tuples should only occur once in the set $P$. Tuples are equal when e.g. $\{a, b\} = ...
3
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3answers
109 views

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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2answers
33 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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1answer
26 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 ...
1
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1answer
24 views

Permutations (Making numbers from digits) [closed]

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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5answers
67 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 ...
0
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1answer
33 views

How would I calculate the total number of combinations [closed]

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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1answer
37 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 ...