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

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3
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0answers
15 views

Determinant of $\delta$ function

Let $$\delta_i^j=\left\{ \begin{aligned} 1 ~~~~~~i=j \\ 0 ~~~~~~i\ne j \end{aligned} \right. $$ $1\le i,j\le n$. How to prove $$ \begin{vmatrix} \delta_{j_1}^{i_1} ~...~ \delta_{j_n}^{i_1} \\ \\ ...
1
vote
0answers
6 views

Calculating the number of permutations that do not have at least one set of duplicate elements adjacent.

Ok, so I've got a set of elements, some are duplicates but each are considered unique as far as set-making goes. I need to find how many permutations exist that do not put two of the duplicates next ...
0
votes
1answer
23 views

Mr and Mrs Zimmerman want to give their baby a first name and a second name so that the baby's three initials are in alphabetical order.

Mr and Mrs Zimmerman want to give their baby a first name and a second name so that the baby's three initials are in alphabetical order. How many different initials could this baby end up with eg. ...
1
vote
1answer
17 views

Permutations of $n$ objects where $r = n -1$

In my text book the question is as follows: Find the way in which $5$ persons can sit in a row if two insist on sitting next to each other. They give the answer as $48$. I fail to understand how ...
1
vote
2answers
29 views

Permutations and Combinations based probblem

Find the value of the expression: $$ 1+1\times1!+2\times2!+3\times3!+.....+n\times n! $$ It is a problem based on the concept of permutations and combinations I don't have a perfect idea to solve ...
8
votes
0answers
74 views

Covering pairs with permutations

Consider an $n \times n$ matrix $M_n$ with the following properties: Each row is a permutation of $A_n \equiv \{1, 2, ..., n\}$. Every ordered pair $(i,j)$, $i,j \in A_n$, $i \neq j$, appears as a ...
-2
votes
1answer
17 views

For any permutation $ \sigma \in S_n$, $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even when $n$ is odd [on hold]

Let σ be a permutation of ${1, 2, 3, . . . , n}$, n odd. I want to show that $(σ(1) − 1)(σ(2) − 2) . . . (σ(n) − n)$ is even. Thank you.
-2
votes
1answer
16 views

Numbers of words allowing repetition [duplicate]

10 different letters of an alphabet are given. words with 5 letters are formed from these given letters.I have to determine the number of words which have at least one letter repeated. Answer is - ...
0
votes
1answer
41 views

Combinatorics problems involving permutations

Let $A= \{ 1,2,3,...,n\}$ a set and $f:A \to A$ a permutation of the set A. We call a number $x \in \{ 2,3,...,n-1 \}$ special if $f(x)>\max \{f(x-1),f(x+1) \}$ or $f(x)<\min \{f(x-1),f(x+1) ...
0
votes
4answers
18 views

sum of numbers formed by permutations

I have digits 2,3,4,5. I have been asked to find the sum of all 4 digits the numbers that can be formed using these digits without repetition such that all are included in the number. Can someone ...
1
vote
1answer
29 views

I need help answering a few simple math problems related to permutations and probability

Question 1: How many words can you make from the letters Texas if repeats are not allowed? Question 2: How many words can you make from the letters Texas if repeats are allowed? Question 3: What is ...
0
votes
0answers
26 views

Divisibility of N [on hold]

If there are $N$ tuples $(a,b,c)$ such that they are positive integers and $a>b>c$ and $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2}$$ then $N$ is divisible by ? The answer is an integer ...
2
votes
1answer
58 views

What is the probability that a five-card poker hand has four ACES?

What is the probability that a five-card poker hand has four ACES? When I was solving the above stated problem, I got confused while trying different methods : Assume a normal $52$ deck of ...
0
votes
1answer
21 views

Polynomial roots conditions vary with coefficients

Polynomial equation $\sum_{i=0}^4 p_i x^i=0$ have the following root conditions: 1) $a_0 \pm b_0 i, a_1 \pm b_1i$ 2) $a_0 \pm b_0 i, a_1, a_2$ 3) $a_0, a_1, a_2 \pm b_2i$ 4) $a_0, a_1, a_2, a_3$ I'm ...
0
votes
2answers
15 views

Sign of composition of transpositions

Let $\sigma \in S_n$. Definition: Suppose that $\text{sign}\sigma=(-1)^N$, where $N$ - number of inversions in permutation $\sigma$. Suppose that $\tau_1$ and $\tau_2$ transpositions. How to prove ...
0
votes
1answer
27 views

Find commuting elements within a permutation group

The question is like this: IF $G=S_5$ and $g=(1\quad 2\quad 3)$, determine the number of elements in $H=\{x\in G:xg=gx\}$. To do the question, first it says $$x(4)=(x(1\quad 2\quad 3))(4)=(1\quad ...
-1
votes
1answer
18 views

How many possible placements are there for a Battleship puzzle?

I am studying the NP-Completeness of the battleship puzzle; the pencil and paper game found in newspapers and not the more popular 2-player version. I understand why the puzzle is NP-Complete because ...
1
vote
1answer
33 views

Fourier transformation of a group

At the beginning of the section 4 of Fast Quantum Fourier Transforms for a Class of Non-abelian Groups, it is said that, ... calculating a Fourier transform for a group $G$ is the same as decomposing ...
1
vote
1answer
17 views

Sign of permutation. Confusing example

Let $\sigma=(2314)\in S_4$. We have different definitions of sign of permutation. 1) Our $\sigma=(24)(21)(23)$ hence $\text{sgn}\sigma=(-1)^3=-1.$ 2) Our $\sigma$ has two inversions namely $(2,1)$ ...
0
votes
1answer
18 views

Does every partition of n correspond to some permutation of [1,2, … n]?

It is known that every permutation can be decomposed into disjoint cycles. The cycle type gives the length of each cycle. The sum of cycles length is n. I am wondering whether every partition of n ...
2
votes
1answer
40 views

what is the smallest non-abelian finite group which has normal, non-abelian subgroups (plural)

I am looking for smallest example of a group $G$ such that: $G$ is a finite, non-abelian group $G$ is not simple $G$ has non-trivial, proper, normal subgroups: $H_1, H_2, \dots $ $H_1, H_2, \dots $ ...
0
votes
0answers
21 views

Cycle structure of the generators of the dihedral group

Would the following be correct about generating the dihedral group $D_n$ by permutations? If $n$ is even, the group can be generated as $\langle(2\quad n)(3 \quad n-1) \ldots (\frac{n}{2}-1 \quad ...
1
vote
2answers
52 views

Partition of natural number not equal to factorial

I wish to prove the following statement so I can use it as a lemma for a group theory result. To be honest I have not tried much yet, my intuition tells me this is going to be connected to the ...
0
votes
1answer
384 views

Determining Permutation/Combinations with Bit Strings

I've got a discrete math problem on my hands...I'm trying to understand the steps behind working with bit strings; specifically, how a bit string of x length has "at least" or "exactly" a certain ...
0
votes
1answer
21 views

permutations of n objects

Does the number of permutations of $n$ objects, $r$ alike of one kind and $n−r$ alike of another kind, always equal the combinations of n different objects taken $r$ at a time? Explain. I know ...
0
votes
1answer
15 views

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 ...
2
votes
1answer
81 views

Random permutations composition

I'm trying to prove a theorem that seems very intuitive. However, I seem to be missing a piece of the puzzle. If: $\pi$ is a random permutation ($S_n$), $\pi_1, \pi_2$ - random permutations with ...
-3
votes
1answer
68 views

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 ...
2
votes
4answers
7k views

how many 5 digit numbers are there with distinct digits?

I found this question on gre forum, it's answer was given by this expression: $9\cdot9\cdot8\cdot7\cdot6$ which I heard in school as well. What I tried to do was: for numbers from index $4$ to ...
7
votes
2answers
549 views

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 ...
3
votes
1answer
30 views

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 ...
0
votes
3answers
71 views

How many distinct ways can the number be written as product of $3$ factors?

How many distinct ways can the number $126$ be written as a product of $3$ positive integer factors? I found that the prime factors are $126=2\times3\times3\times7$. But how to get number of ...
2
votes
1answer
19 views

For each of the following restrictions, find the smallest size n for strings over $\{a, b, c\}$ that can be used as codes for $27$ people.

For each of the following restrictions, find the smallest size $n$ for strings over $\{a, b, c\}$ that can be used as codes for $27$ people. a. There are $k$ $a$’s, $l$ $b$’s, and $m$ $c$’s and $k + ...
3
votes
2answers
41 views

What is probability that out of the first half on N objects, none will be matched with their own label?

The problem: We have N (even) objects ordered $o_1 ... o_N$ , each having their own label. The labels are reassigned to the objects randomly. What is the probability that that neither of the first ...
5
votes
2answers
95 views

If we are given that a list of $n$ numbers has $11,660$ derangements, what is the value of $n$?

The Full Question For the positive integers $1,2,3,\dots n-1,n$, there are $11,660$ where $1,2,3,4,5$ appear in the first five positions. What is the value of $n$? My Work First I considered all ...
2
votes
1answer
22 views

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 ...
0
votes
0answers
23 views

Centralizer of $\sigma\in S_n$ [duplicate]

Let $\sigma\in S_n$ . Describe the centralizer of $\sigma$. Thought: If I conjugate $\sigma$, then $\tau^{-1}\sigma\tau=\sigma.$ This means that $\tau$ is a power of $\sigma,$ and ...
1
vote
1answer
24 views

How many ways are there to arrange the letters of word $ALGEBRA$ such that the relative order of the vowels and consonants doesn't change?

I did this question this way :- there are 4 consonants in the words (LGBR) and there are 7 letters in the word. $therefore$ number of in which consonants can be arranged in relative order will be ...
0
votes
2answers
35 views

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 ...
0
votes
4answers
35 views

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 ...
3
votes
3answers
272 views

Which Digit-Permutations Preserve Divisibility?

This is a completely random question that just happened to come to mind recently and I was wondering if the MathSE community had anything to say about it. Let $n > 1,b > 1$ be integers and ...
2
votes
1answer
61 views

Number of permutations in lawn tennis so no husband and wife play together.

In how many ways can a lawn tennis mixed doubles be made up from seven married couples if no husband and wife play in the same set? Please explain the logic.
4
votes
3answers
77 views

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 ...
1
vote
2answers
49 views

Counting the number of “distinct” permutations of two sets?

I don't really know how to introduce this question, so I start defining something I needed in order to well understand the problem I met! Let $A$, $B$ two finite sets of distinct elements, with ...
3
votes
2answers
63 views

math software - permutation group elements operation

I need a software allows to calculate operation elements of permutation group. For example the following elements operation yields identity permutation $$ (1234)(1423) = (1)$$ Sage seems to solve the ...
1
vote
2answers
20 views

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 ...
1
vote
1answer
31 views

rationale for book's solution of combinatorics question about scheduling ten speakers with restrictions

If A, B, C are among $10$ people speaking at a function in alphabetical order What are total ways of doing so. BOOKS APPROACH: There are $10$ people out of which $3$ need to be taken care of. So ...
0
votes
3answers
43 views

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A?

In how many ways can we arrange the letters of word BAHAMA such that it starts with H and ends with A? I have a doubt in the selection of A at the last position. Please help. Thanks in advance!!
0
votes
1answer
69 views

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 ...
0
votes
1answer
31 views

Permutations of 3 digit numbers divisible by 5

I recently had to answer the following permutation question: How many 3 digit numbers can be formed from the digits 2,3,5,6,7,9 which are divisible by 5 and none of the digits are repeated? ...