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Questions tagged [permutations]

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

The word permutation has several possible meanings, based on context. In combinatorics, a permutation is generally taken to be a sequence containing every element from a finite set exactly once. Permutations of a finite set can be thought of as exactly the ways in which the elements of the set can be ordered.

In group theory, a permutation of a (not necessarily finite) set $S$ is a bijection $\sigma : S \to S$. The set of all permutations of $S$ forms a group under composition, called the symmetric group on $S$.

Reference: Permutation.

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Prove that for $n>0$ number of permutations of set $\{1,...,n\}$ for every $i=1,2,..,n-1$ is equal to....

Prove that for $n>0$ number of permutations of set $\{1,...,n\}$ such that $a_{i+1}-a_i \neq 1$ for every $i=1,2,..,n-1$ is equal to $$D_n + (n-1)D_{n-2} + (-1)^{n-1} $$ where $D_n$ is number of disorders. Could you give me a hint ?
xawey
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Number of ways to form two different committees

There are five employees willing to serve on one of two different committees. If each employee can only serve on one committee, how many possible ways are there for the openings on the committees to be filled? The answer is 20. And the…
Quaxton Hale
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Bijection from permutations to restricted n-tuples

Let $A_n$ be the set of permutations of $1,\dots,n$. Let $B_n$ be the set of $n$-tuples $(b_1,\dots,b_n)$ such that $1\le b_i\le i$ for each $i\in 1,\dots, n$. Construct a bijection from $A_n$ to $B_n$. (Hint: Use induction on $n$, employing…
user782220
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Arrangement of 4 boys and 4 girls with no two girls are next to each other?

Let we have 4 boys and 4 girls and we need to arrange them, with no two boys and no two girls are next to each other? Let place first the boys b b b b the 4 boys can arrange themselves by 4! ways. Now we have 5 positions between boys but we have …
user52950
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Mapping permutations with repetitions to an index

I am looking for a way to map permutation with repetitions written in big array (for example with million elemenths) to an index written in other array. Every element in array can have values from 0 to 255. How to achieve that? Example (not…
maszynaz
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A problem on counting

Consider a number system which does not have the number 7 but has all other numbers. So the numbers are $1,2,3,4,5,6,8,9,10,11,12,13,14,15,16,18,...$. I want to find what is the $10^k$ number, where $k$ is an integer. For example $10^{th}$ number is…
happymath
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Maximum winner matches

N players take part in tennis championship. In every match loser is out. Two players can play a game if in that moment the difference of played games of that two players is not more than 1. They are interested, how many matches will be in the…
savvi singh
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Permutation that gives a sequence of non-negative partial sums

Given a sequence $(a_i)_1^n$ of real numbers that sum to $0$. There are $n$ circular permutations. $\begin{array}{}\sigma_1 = &(a_1 &a_2 &\cdots &a_{n-1} &a_n)\\ \sigma_2 = &(a_2 &a_3 & \cdots &a_n &a_1)\\\sigma_3 = &(a_3& a_4& \cdots &a_1…
Anant
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Security of permutation function

I have a question related to permutation. If I have a 10-bit number: {0101110010}. I want to swap even position bits to adjacent odd bits and odd positions to adjacent even bits, such that I get {1010110001}. Can I call it an alternate…
Alexandera
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Permutation and combinations count of members per group

A club with $x$ members is organized into four committees such that (1) each member is in exactly two committees, (2) any two committees have exactly one member in common. Then $x$ has (A) exactly two values both between $4$ and $8$ (B) exactly one…
square_one
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Permutation & Combination Problem

I often solve math questions because I like it (This may sound crazy, I know :)). Today I came across an interesting permutation & combination question. The question is as follows: 6 people (named A, B, C, D, E, F) are in a line in a supermarket.…
Rakanoth
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My proof regarding composition of permutations came to the same conclusion as the answer sheet, but through different methods. Is it valid?

Let $S_3$ be a set of all permutations of elements in $\{1,2,3\}$. Prove that there doesn't exist f $\in S_3$ where $\{f,f^2,f^3,f^4,f^5,f^6\} = S_3$. Where $f^n = f \circ f \circ \:... \circ \:f$ $n$ times. The proof in the answer sheet goes about…
Jake Browning
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Permutations of two photo frames

Please help with this permutations question. I'm trying to use the permutation formula to calculate it but don't know where to begin: $$\frac{n!}{(n-r)!}$$ Here's the problem: Jane is choosing photos to display in 2 frames. Each frame holds 4…
Alex
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How many ways are there to place these books on the shelves?

You are given 5 books and 7 bookshelves. How many ways are there to place these books on the shelves? (The order on the shelves matters.) I want to say $7^5$ since there are 7 possible shelves and five different options to select from books that…
GivenPie
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Need a counterexample: A product of a cycle with itself will in general be a cycle.

Need a counterexample: A product of a cycle with itself will in general be a cycle. I was thinking something in S_4.
Guest27
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