# Tagged Questions

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

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### Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given to ...
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### Simple Question on Binomial theorems… [closed]

I have tried to solve this question by putting the value of each coefficients but it is really becoming very lengthy.... what i got was this number 10510100501... But how to get this in the required ...
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### Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...
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### Understanding derangement.

From the inclusion-exclusion principle we get that out of $N$ objects with one label each, there is a probability of $$\sum_{k=1}^N (-1)^{k+1}\frac{1}{k!}$$ that a random assignment of the $N$ labels ...
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### How many straight lines can be made between 10 points such that 4 of them are colinear?

So i know how to get the answer. We just have to find $C(10,2)$ and subtract $C(4,2)$ and add 1. We are basically counting all the points between co-linear points as 1. So the question is why we are ...
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### 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 ...
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### Finding number of functions from a set to itself such that $f(f(x)) = x$

The questions states that $f: A\rightarrow A$ is a function which satisfies $f(f(x)) = x.$ We have to find the number of such functions with $A = \left\{1,2,3,4\right\}$. The given condition clearly ...
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### A question of permutations and combinations with six cards and six envelopes.

Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same ...
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### How do I calculate the number of unique permutations in a list with repeated elements? [duplicate]

I know that I can get the number of permutations of items in a list without repetition using (n!) How would I calculate the number of unique permutations when a ...
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### Painting a 2x2 Grid

We have a 2x2 grid and 10 different colours. I want to paint such that adjacent grids are painted with different colors. How many ways can i do this? ...
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### $5$ chem students, $6$ maths students and $7$ physics students permutation

$5$ chem students, $6$ maths students and $7$ physics students. Find the number of arrangements if a)Chem majors are to occupy the first 5 positions b)Chem majors cannot occupy the first 5 positions ...
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### Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph

I am trying to understand Conway's theorem on the number of orbits on the set of all ordered cycles in a $d$-valent graph. I quote it from Cycles in graphs and groups by Kantor. Theorem $1$ (...
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### Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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### Finding the parity of a permutation “exclusively”?

I'm trying to find the parity of permutations such as $(2468)$. What makes it possible to find the "exclusive" parity of such permutation? I.e. that if one tries to express $(2468)$ as a product of ...
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### If $\sigma=(a_1 a_2 … a_n)$ and $|\sigma|$ is odd, then what is $\sigma^2$?

I'm trying to understand the way to infer the power of a permutation. If $\sigma=(a_1 a_2 ... a_n)$ is a $k$-cycle and $k=|\sigma|$ is odd, then how can I infer what $\sigma^2$ is?
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### Prove $sgn(π) = sgn(π^{-1})$?

I'm pretty sure the inversion count of $π$ should be the opposite of the inversion count of $π^{-1}$. By this I mean if $π$ looks like this: $1 \to 1$, $2\to 2, \ldots, 10 \to 10$ and therefore the ...
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### Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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### Write $π = (3, 2, 5)(2, 5, 4)$ in “table” notation?

Isn't this impossible...? Because this permutation goes from 3 --> 2 ---> 5 ---> 3 according to the first cycle, but goes from 2 --> 5 ---> 4 ---> 2 according to the second cycle. So 5 can't go to 3 ...
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### Permutation of the alphabets of the word “mediterranean” such that first and fourth letter are “r” and “e” respectively.

Above is the original question. The correct answer is in green that is 59. I have chosen option 3 that is $\frac{11!}{(2!)^3}$ because I thought that there are 13 alphabets in the word "mediterranean"...
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### How can we compute restrictions from a character table?

I would like to how to, when given a character table, calculate the restriction. $Res_H^G : Rep(G) \rightarrow Rep(H)$. For example: Let $G=S_4$ whose character table is given below (see ...
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### Need help calculating number of possible passwords with given criteria

I need help calculating the number of possible passwords with a given set of criteria. Here is the set of criteria: Passwords are case insensitive. Must be 6-14 characters. Must contain at least 1 ...
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### Counting monomials with $k$ variables

Say we expand $\left(\sum_{i=1}^n x_i\right)^k$ into monomials. If $k=3$ there are $3n(n-1)$ monomials with two variables: $3x_1x_2^2 + 3x_1x_3^2 +\dots + 3x_1^2x_2 + \dots$. Is there a closed form ...
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### Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$.

$S_n$ is the set of all permutations. Show that $(στ)^{-1} = τ^{-1}σ^{-1}$ for all $σ, τ ∈ S_n$ I can somewhat see why this statement would be true, seeing as permutations are read from right to ...
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### Suppose $π ∈ S_n$, and for this $π$ define $C_π : S_n → S_n$ be defined by $C_π(σ) = πσ$. Why is $C_π$ a bijection?

$S_n$ is the set of all permutations. I'm just starting on this material, so I'm confused on how to read this problem. Does the function consist of multiple permutations (i.e. the permutation of a ...
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### Find The Number Of Outcomes

I understand how to find the number of outcomes using permutations and combinations, but then I thought to myself what happens when it involves both? I will make a mock scenario to explain what I am ...
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### the length of the conjugate class containing $\alpha$ in $S_n$ [duplicate]

Suppose $\alpha$ $\in$ $S_n$ and there are exactly $n_i$ $l_i$-cycles ($i=1,2, ... ,k)$ (containing $1$-cycles) in the cycle decompostion of $\alpha$ ., then the length of the conjugate class ...
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### Converting Permutations to Combinations: Simple Stats in Practise

In a popular text book there is a question that has bothered me that I am sure is very simple for others and I'm just missing something..... So image $100$ songs and we have $10$ as Beatles songs. We ...
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### Generate a unique combination from an index within the number of combinations

I'm writing a program which will use a genetic algorithm optimize neural networks to play tic-tac-toe (That's not related), and I've come across the following problem: I'm looping through every ...
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### Permutation : Is there any formula to solve this?

Given, 14 objects of type A 8 objects of type B 3 objects of type C 2 objects of type D Find the permutation of 10 objects? Is there any general formula in permutation to solve a problem ...
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### What is an intuitive explanation of the combinations formula?

I perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And ...
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### Is $N_{A_7}(H) = H$, with the following $H$?

I am following a proof in which I have a subgroup of $S_7$ defined by $H := \langle (2, 3, 4)(5, 6, 7) , (2, 7, 6, 3)(4, 5) \rangle$ The book implicitly uses that $N_{A_7}(H) = H$ (the normalizer ...