Questions tagged [permutations]
For questions related to permutations, which can be viewed as re-ordering a collection of objects.
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Number of ways to arrange 0 marbles?
A game has to be made from marbles of n colors, where n marbles has to be kept one upon another. In how many ways these marbles can be arranged?
Is the answer is $1$ if $n = 0?$
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1answer
14 views
Find permutations from their composition
$f$ and $g$ are permutations such that
$f \cdot f = id,$
$g \cdot g = id,$
$f \cdot g = \left( \begin{array}{cc}
1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
10 &...
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1answer
16 views
Permutations & Combinations: Probability & Statistics [on hold]
thank you for stopping by and thanks in advance for your time! I'm struggling with a probability & statistics problem while studying for my exam so any help would be much appreciated and even more ...
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2answers
20 views
Discrete math counting principles
In how many ways can a teacher distribute 12 identical science books among 15 students if
1) no student gets more than one book?
2) if the oldest student gets two books but no other student ...
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1answer
23 views
Permutations vs Combinations: Probability & Statistics
I need help with the following exercise as I am preparing for my exams on my own and never had the chance to visit any workshops or lectures as they were not offered, so here it goes.
A lock can be ...
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0answers
34 views
In how many ways can $3$ boys and $2$ girls sit in a row, where none of the 3 boys can sit next to an another boy?
As I see it, the row can only be formed in that way : $BGBGB$ which means that there are $3!\cdot2! = 12$ ways but my book says that the answer is $24$. Am I missing something ?
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0answers
22 views
show that the order of C(a) must be odd
I am working on the following problem from group theory:
If $n$ is odd and $a\in S_n$ is an n-cycle, $a=(a_1,a_2,......,a_n)$, show that no element of the centralizer $C(a)=\{g\in S_n \mid ga=ag\}$ ...
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1answer
17 views
How many strings can be formed by ordering the letters ABCDEF such that the string contains neither of the substrings AD or BEF? [on hold]
Im really lost on how to do this/approach since my professor rushed this section in class. Any help will do please!
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2answers
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Letter permutation MISSISISPPI, S come before any I
If there is no restriction, the number of ways to organize letter of MISSISSIPPI is,
$$ \frac{11!}{4!4!2!} $$
The restriction is,
all Ss come before any Is. So I group both letters, then there are ...
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2answers
70 views
Is there an $11$-element circular permutation of $\{1,2,…,12\}$ with all $|a_i-a_{i+1}|$ distinct?
Can you choose $11$ different numbers among them so that the numbers $|a_1-a_2|, |a_2-a_3|,\ldots,|a_{10}-a_{11}|,|a_{11}-a_{1}|$ are all different. The smartest thing that my dumbest mind could ...
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1answer
34 views
Show that $\alpha = (1 2 3)(2 3 4)(5 6 7)(7 8 9 10)$ has order $10$ in $S_n$ $(n\geq10)$.
So I have done problem #13 in section 7.5 of t book Abstract Algebra: An Intorduction, 3rd Edition by Thomas W. Hungerford (ISBN: 978-1-1115696-2-4). multiple times over now, but I still get the order ...
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1answer
25 views
How many random 3 letter words have exactly 1 “A”? No repetitions. Trying to build intuition here
I think the options are either:
c(3,1) * c(25,2) (as in, choose 1 of the three letters to be the “A” and choose 2 of the remaining 25 letters for the other one) = 900
OR
if the first letter is an A,...
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vote
1answer
41 views
How to compute the powers of $\tau$?
When working through the Symmetric and Alternating Groups section of my Abstract Algebra textbook, I came across Theorem 7.25 that sates:
The order of a permutation $\tau$ in $S_n$ is the least ...
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0answers
18 views
How many strings of four decimal digits do not contain the same digit twice
The answer should be
1: Four different digits : $$10*9*8*7 = 5040 $$
2: Four different digits + contains the same digit three times + all digits are of the same : $$ 10*9*8*7+10*{4 \choose 3} *9 + ...
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3answers
41 views
How many different arrangements are there if Bob and Sally must always be seated next to each other?
How many different arrangements are there in which Bob, Sally and $n$ other people sit down in a row of $n+3$ chairs if Bob and Sally must always be seated next to each other?
I tried putting Bob ...
4
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1answer
40 views
Four men go into a restaurant and leave their umbrellas at the door
Question:
Four men go into a restaurant and leave their umbrellas at the door. On their way out, each man picks up an umbrella and they discover when they get outside that no man has his own ...
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0answers
31 views
tiling problem - 1x 6 board
I have 4 colours to choose from and have a board that is 1x6. How many tilings can I make with no colour repeating more than twice? How many tilings can I make with no more than one colour being ...
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0answers
33 views
Counting Logic for Subsets, Password Combination, etc. Need clarification if my understanding is correct to approach these scenarios?
I have a few different types of counting problems where I feel I don't understand the approach to specific counting questions. I will list down the different scenarios with how I logically see it.
...
1
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1answer
25 views
The lobster eating problem, a circular table and the possible table arrangement
The following problem was a fun combinatorics question I encountered, there seems to be a buildup in the question, I wanted to ask if my reasoning sounds plausible. I'm sorry if my reasoning is a bit ...
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1answer
43 views
How many combinations of coins add up to \$20
We have four coins
Coin 1: $0.10
Coin 2: $1.00
Coin 3: $1.00
Coin 4: $1.00
How many ways can we get $20.00 from these coins?
My attempt:
I started by counting the total number of ways for each ...
2
votes
2answers
75 views
let $A=\{1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5\}$. Number of one-one function from $A$ to $B$
Let $A=\{1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5\}$. Number of one-one function from $A$ to $B$ such that $f(1) \neq 0$ and $f(i)\neq i$ for $i={1,2,3,4,5}$ is _______ .
So I know one one function means ...
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1answer
25 views
Recursion $I_{n+1} = I_n + nI_{n-1}$ for the amount $I_n$ of the involutions in $S_n$
A permutation $\pi \in S_n$ is an involution, when $\pi^2 = \text{id}$.
How can one show for the amount $I_n$ of the involutions in $S_n$ the following recursion:
$$I_{n+1} = I_n + nI_{n-1}$$
...
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0answers
27 views
Can you show me the way? [on hold]
Suppose you secured your bike using a combination lock.Later, you realized that you forgot the 4-digit code.You only remembered that the code contains the digits 1,3,4 and 7.
2
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1answer
26 views
Permutation group of Cosets
Let $K$ be a subgroup of a group $G$. Let $T$ denote the set of all distinct right cosets of $K$ in $G$ and $A(T)$ be the permutation group of $T$. Prove the following statements.
(a) For each $a\in ...
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1answer
18 views
Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.
Compute all $X_{g}$ and all $G_{x}$ for $X = \left \{1, 2, 3\right \}$, $G = S_{3} = \left \{(1), (12), (13), (23), (123), (132)\right \}$.
Can someone give me a head start to this problem?
$X_{g}=\...
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2answers
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Cycle structure of the permutation $x \mapsto p·x\operatorname{mod}q$ for coprime $p,q$
Let $[q] = \{0,\dots,q-1\}$, $p < q$.
Consider the function $\mathbf{p}: [q] \rightarrow [q]$ which sends $x \mapsto p·x\operatorname{mod}q$, i.e. the multiplication by $p$ modulo $q$ on $[q]$.
...
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2answers
33 views
Number of ways to arrange the letters BANANAS
I was in class, and the teacher told us, that the number of ways to re-arrange the letters in BANANAS is $\frac{7!}{2!3!)}$. I understand where the $2, 3$ and $7$ are coming from (the number of $N$'s, ...
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2answers
21 views
7 reindeer in a single file line, dasher cannot be next to prancer.
overall, I can see that there would be 7! arrangements, but I don't know how to remove the instances where Dasher is next to Prancer. I can think of 12 arrangements of which dasher is next to prancer.
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0answers
30 views
Presentations of $D_8$ using permutations
I fond a lot of examples using presentation of $D_8$ by generators which are permutations of $S_4$.
1) How many presentations could be found?
2) Could it be presented by permutations of $S_5$ or ...
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1answer
40 views
Show that each permutation in $A_4$ has a square root.
Show that $ A_4 = \{ \sigma \in S_4 \mid \sigma = \tau^2 \text{ for some }
\tau \in S_4 \} $.
$S_4$ is the permutations of 1,2,3,4.
$A_4$ is the alternating group of $S_4$.
Let $ B = \...
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2answers
30 views
Need help simplifying with factorials [on hold]
I'm doing permutations and I need help with the equation:
$(n+1)!=6(n-1)!$
Please explain all the steps on how to complete this question as I am very confused! Thanks.
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2answers
36 views
A nonidentity permutation $\sigma$ satisfies $\sigma(i)<i$ for some $i$ [closed]
If a permutation $\sigma: N\to N$ is not the identity, prove that there exists an $i \in \{1,...,n\} $ such that $\sigma(i)<i$.
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1answer
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circular r-permutations of n
My book, Discrete Mathematics And Its Applications by K.H.Rosen asks to find a formula for circular r-permutations of n people. That is, sitting of r of these people around a table when two sittings ...
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0answers
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Inversion number that form a set
For $n≥3$ determine the set
{${\sigma∈S_n| |inv(\sigma)|=\frac{(n+1)(n-2)}{2}}$}
Now I know that in general $0≤|inv(\sigma)|≤\frac{n(n-1)}{2}$ for $\sigma∈S_n$
What do I have to consider? Sorry ...
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1answer
19 views
Filling a barrel, using small containers
In how many ways, using containers, one with 2 liters and other with 7 liters, can you fill a barrel of 1234 liters?
What's the fastest and what's the slowest way to fill the barrel?
Should I use ...
1
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1answer
31 views
Number of ways to organizing n object types into 2n slots, requiring each type to be in 2 slots
For example, say I have $3$ objects, $6$ slots, and each object must be chosen twice, how do I go about solving that?
Would it just be $\binom{6}{2} \binom{4}{2} \binom{2}{2}$ or am I thinking about ...
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1answer
22 views
Three persons entering a railway carriage where there are 5 vacant seats. In how many ways can they seat themselves?
In the question, three persons entering a railway carriage where there are $5$ vacant
seats. In how many ways can they seat themselves?
Why are we multiplying $5\times 4\times 3$? Ive read the ...
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1answer
26 views
Probability of overlap of random intervals dropped on unit circle
Suppose I have a circle with circumference $A$. Along the circumference of this circle, I randomly drop $N$ arcs with fixed length $a < A$. Now suppose I drop a single additional arc ($N+1$). What ...
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1answer
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Higher Order Multivariable Taylor Expansions
The quadratic multivariable Taylor approximation of a function $f(x, y)$ around a point $(a, b)$ is given by $f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) + \frac{1}{2}f_{xx}(a, b)(x - a)^2 + f_{xy}(...
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1answer
26 views
What is the probability that only one of the cards will have the matching suit?
A group of 4 friends are playing a card game using a standard deck of 52 cards. Each friend receives one card from the deck, and the next card is flipped up. Only one of the friend has a card that ...
2
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1answer
95 views
How many ways can I arrange 5 As, 6 Bs, and 3 Cs [duplicate]
With requirement that $A$ precedes the first $B$ which precedes the first $C$.
Example: $AABABCBBAACBCB$ is correct and $BAAACBAACBCBBBB$ is incorrect.
I am thinking of using permutation, get the ...
1
vote
1answer
35 views
Permutations and terminology
Say I have the following permutation
$$\sigma ={\begin{pmatrix}1&2&3&4&5&6&7&8\\1&2&3&8&4&5&6&7\end{pmatrix}}$$
which consists to let ...
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1answer
29 views
Expected Value for uniform random permutations
Question:
Let $n\geq 2$ be an integer and let $a_1, a_2, ..., a_n$ be a uniformly random permutation of the set $\{1,2,...n\}$. Let $X$ be the random variable with value:
$X$ = the number of ...
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0answers
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Properties of sets of permutations
This is a very vague question and not particularly related to anything in general, but any thoughts would be helpful.
Say you have permutations $\pi_1,\dots,\pi_n$ of $[2n+1]$. What are some ...
3
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0answers
79 views
Why are permutations (nPr) called variations in non-English languages?
First of all, you should be at least a little familiar with combinatorics to understand that question.
Some often used calculator keys in stochastic are the nCr and nPr ones.
Edit: I've first asked ...
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0answers
24 views
How many bit strings of length $8$ have $3$ x's in a row?
A strings of length $8$ using the letters x and y only
1) How many of these strings have exactly $3$ x's? So it's $\binom{8}{3} = 56$
2) How many of these strings have 3 x's in a row?
I don't know ...
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0answers
11 views
Permutations and Combinations involving words
How many words can be formed by taking 4 letters at a time from the letters of the word: MORADABAD
I am getting confused as to the several cases possible in this particular word. Kindly assist in a ...
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1answer
22 views
On a grid of m × n squares, how many ways exist to get from the top-left corner to the bottom-right corner? [closed]
You can only move right or down on an edge.
0
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1answer
11 views
Discrete Probability: Uniformly random subsets, permutations and birthday probability
Question a)
Let $n$ $\ge $ $2$ be the number of students who are writing an exam. Each of these students has a uniformly random birthday, which is independent of the birthdays of the other ...
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0answers
23 views
Is $\sum_{k=0}^{n} \left({n \choose k}\right)^{2}$ equal to ${2n\choose n}$? [duplicate]
Substituting some numerical values for $n$ leads me to believe that $\sum_{k=0}^{n} \left({n \choose k}\right)^{2} = {2n\choose n}$, but how would I show that, if it's indeed true?
