# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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 $B$ ...
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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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. ...
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### 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 ...
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### 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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### 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 ...
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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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### 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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### 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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### 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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### 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 ...