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

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A combinatorics question about selection strategies

I am given a set of balls--red and blue. In each set, there are three kinds of balls--small, medium and large. In each set there are 10 balls of each color: 10 Red balls (2 small + 3 medium + 5 ...
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1answer
16 views

How do I read this equation related to Combinations with repetitions in natural language?

Here's an Article from TopCoder about Combinatorics, that starts by introducing some basic concepts such as: Combinations and permutations. That part I understood just fine, but then the article ...
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0answers
19 views

Bit String Probability [on hold]

Given a bit string of length 8 begins with a 0, find the probability that it contains exactly three 0's. How many bit strings of length 8 contain an evan number of 0's? How can permutations and ...
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1answer
20 views

Bitstring Probability

I am not understanding how to apply n choose r and permutations to the following problem. Given a bit string of length 8 that has exactly three 0's, what is the probability that the bit string will ...
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2answers
28 views

How many 5 digit numbers can be formed out of {1,2,3…,9} where a digit can repeat at most twice?

The question is: How many different numbers of 5 digits can be generated out of {1,2,3,4,5,6,7,8,9} such that no digit can appear more than twice ? That is a number like 11213 is not allowed. but ...
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0answers
8 views

Prove a certain property of the Hodge double star operator

I want to solve the following problem Show that $\ast\ast\omega = (-1)^{k(n-k)}\omega$ where $ \displaystyle \ast\omega =\sum_I \text{sgn}(I,J)\omega_I dx^J$ and $\omega$ is a k-form in ...
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1answer
19 views

Group theory disjoint cycles

Let $a=(1 3 5)(1 5 6)(1 3 5)$ I had to write this as a product of disjoint cycles and got $(1 5)(3 6)$ which I believe is correct. Then figure out $a^{24}$ and $a^{25}$. Now $a^{24}$ is the ...
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0answers
26 views

Number of Unique Permutations of 3 digits (-1,0,1) given a length that match a sum

Say you have a vertical game board of n length (length being number of spaces). And you have a three sided die that has the options: go forward one, go back one, and stay. If you go below or above ...
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1answer
27 views

Ways to arrange ALGEBRA so AA occurs

So the permutations of this qould be 7!, and I know that there are 2 objects of type A, but how can we isolate the events where those objects occur consecutively?
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1answer
11 views

How to find the rank of linear permutation when replacement is allowed?

Question: If all $5*5*5*5*5*5*5*5=5^8=390625$ 8-digit numbers obtained by arranging (permuting) the five digits $2, 3, 6, 7$ & $9$ with their replacements are arranged in the correct increasing ...
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3answers
31 views

How many different possible expressions can I have?

I have three numbers $a,b$ and $c$ How many different additions can I have ? $a + a + a = 3a$ $a + a + b = 2a + b$ However, $a + b + a =2a + b$ which is the same addition as above so I neglect it. ...
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2answers
15 views

finding number of triangles inscribed in a circle

How to find the number of acute angle and obtuse angled triangles that can be inscribed in a circle containing 'N' equally spaced points.
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0answers
19 views

Order of two vectors to maximise the norm

Given vectors ${\bf a} = [a_1, \dots , a_n]^T$ and ${\bf b} = [b_1, \dots , b_n]^T$, a permutation $\pi$ acting on $[1, \dots ,n]$ and defining ${\bf b}^{\pi} = [b_{\pi(1)}, \dots , b_{\pi(n)}]^T$, ...
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0answers
17 views

How to fill number of positions with given operators? [on hold]

We have 4 position between 5 numbers ....and 3 operators (+,*,/) to fill this position... for example 1_2_10_15_25 we can have 1+2*10*15/25 or 1+2+10+15+25 (Repetition of any operator is allowed) ...
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1answer
33 views

How to multiply permutations together?

This is straight from an exam question: Find the order of the permutation $(1465732)(358)(79)$ in $S_9$ So I understand that I first have to write this permutation in disjoint cycle notation, but I'm ...
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0answers
27 views

Is the sequence generated by two permutations periodic?

It's quite easy to prove that given an application: $\sigma:[1,n]\to [1,n]$ we know that the sequence: $$Id_n,\sigma,\sigma^2,\sigma^3,\cdots,\sigma^m,\cdots $$ Is periodic after some index $k\leq ...
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4answers
42 views

Combinations and Permutations in coin tossing

I understand the formulae for combinations and permutations and that for the binomial distribution. However, I'm confused about their application to coin tossing. Consider three tosses. Outcomes ...
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0answers
34 views

Combinations or permutations

I have 3 particles and 5 energy levels (0E,1E,2E,3E,4E). I require all possible ways such that the sum of 3 particles equals 6E. Is there a formula that would enable me to compute the possible ways?
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2answers
34 views

Combinations and Permutations - tiling a $52\times 3$ grid with $78$ dominos

A grid with $3$ rows and $52$ columns is tiled with $78$ identical $2\cdot1$ dominoes. In how many ways can this be done such that exactly two of the dominoes are vertical. Is this right?- ${78 ...
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1answer
43 views

The greatest number of points of intersection of n circles and m straight lines is-

The question is about combinatorics. I have no idea on how to start solving the problem. Please guide me. $(a) 2mn+ {m \choose 2}$ $(b) \frac{1}{2}m(m-1)+n(2m+n-1)$ $(c) {m \choose 2}+2({n \choose ...
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0answers
18 views
2
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1answer
28 views

Show $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$

Let $n \geq 2$ be an integer and $i, j \in \{1, 2, ..., n\} $ be distinct elements. Let $\sigma \in S_n$, Show that $\sigma^{-1} (i j)\sigma = ((i)\sigma (j)\sigma)$ let ...
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1answer
37 views

2x2 grid game problem

A friend of mine is attempting to make a webpage that has a game for a 2x2 grid that is similar to the old North, South, East, West game. I cannot for the life of me figure this out. Essentially, ...
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1answer
17 views

What would the expected number of swaps in a merge sort be?

If I were given a list of random numbers say x1, x2, .........., xn and these numbers are sorted according to the merge sort algorithm. What would be the number of expected swaps/exchanges which would ...
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2answers
24 views

Permutations; group of 5 boys, 10 girls. What's the probability the person the 4th position is a boy?

Problem description: A group of 5 boys and 10 girls is lined up in random order -- that is, each of the 15! permutations is assumed to be equally likely. What is the probability that the person in ...
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2answers
24 views

If $m$ and $n$ women are standing toghter ]such that no men are woman are adjacent together what are the number of Permutations

suppose $m$ men and $n$ women from a single line in such a way that no two men are next to each other and no women are next to each other how many lineup are possible ? Never solved these problems ...
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1answer
41 views

Generating function of derangements

I am pretty new to the topic of generating functions and I would appreciate if someone could help me out with this problem I have. In the lecture we have proven the following generating function for ...
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0answers
22 views

What is the difference between the middle factor and the middle term of permutation ? [duplicate]

What is the difference between the middle factor and the middle term of permutation ?
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0answers
20 views

Possible combinations of foods [closed]

If I have a salad bar offering 10 dressings, 3 lettuce mixes, 29 toppings how many combinations can I make? Each combination having at least one kind of the three lettuce mixes and one dressing?
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0answers
43 views

what is the middle term and the middle factor of 15P7? [closed]

what is the middle term and the middle factor of 15P7 ?
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0answers
28 views

Combinatorics: Password consisting of 13 characters. Must contain at least one odd digit, and at most two even digits. How many passwords?

I'm really trying here. I just need help where to go next. Each character is one of the 10 digits 0, 1, 2, ... , 9 What I have so far is that there are 10^13 possible passwords. I'd have to subtract ...
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0answers
10 views

Finding a permutation class that has a growth rate greater than 1 and less than 0?

In a permutation class, there is an upper growth rate such that $gr(C)=\limsup_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$ and a lower growth rate such that $\liminf_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$. ...
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1answer
25 views

What did I do wrong in the permutations question.

I was given the following question: A hardware store sells numerals for house numbers. It has large quantities of the numerals 3, 5, and 8 but no other numerals. How many different house numbers, ...
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0answers
26 views

Equivalence of right and left cosets of two different subgroups.

Let $A$ and $B$ be two (not necessarily equal) abelian subgroups of $S_5$. If $x$ is an element of $S_5$, under what condition is the following satisfied $$xA = Bx$$ Update: The original question I ...
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1answer
32 views

Number of ways a multiple choice exam can be answered if no two consecutive answers are the same

How many different ways can you answer a 7- question multiple choice exam (with 3 choices) if you know that no two consecutive answers are the same?
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2answers
61 views

How many sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels?

How many sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels? I am so lost and confused, but here's my approach: ...
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0answers
18 views

Middle term and middle factor of Permutation

How to get the middle term and the middle factor of a permutation (ex: 15P7)? Also what is the difference between them?
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2answers
21 views

Is there a shortcut to this combination problem?

The question I have encountered is: From 4 oranges, 3 bananas and 2 apples, how many selections of 5 pieces of fruit can be made, taking at least 1 of each kind? So the method I used to solve this ...
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0answers
21 views

Possible Permutations of Words [closed]

I have $n$ different words, $k$ zeros, how many possible different strings can they form? For example, I have one word $a$, two $0$, the possible combinations are: $00a,0a0,a00.$ Any number $(\leq ...
2
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1answer
23 views

Possibilities of license plates with special rules

I have looked all over the web for some additional information on this matter with no results. Lets say a new form of license plate have 4 letters followed by 3 digits and all sequences are possible. ...
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1answer
14 views

Inversions and Multiplicativity of the Sign of a Permutation

The question is mainly about showing, for two permutations $\sigma, \pi \in S_{n}$, that $\mathrm{sgn}(\sigma \pi) = \mathrm{sgn}(\sigma) \mathrm{sgn}(\pi)$ using inversions of permutations (i.e. a ...
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36 views

Let $f \in 8^8$ where the permutation is given in two line form:

I'm having trouble understanding how one would answer question's like this. $$\begin{pmatrix} 1&2&3&4&5&6&7&8\\ 3&4&5&8&1&2&7&6\end{pmatrix}$$ ...
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2answers
41 views

Finding probability of intersection of events

I was reading First course in Probability by Sheldon Ross and am stuck at the understanding this simple problem [hence proved my maths is poor :( ]. Problem: Celine is undecided as to whether to ...
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4answers
121 views

Number of ways you can form pairs with a group of people when certain people cannot be paired with each other.

Let's say you have a group of eight people and you want to form them into pairs for group projects. There are $\frac{8!}{4!.2!}$ ways to do it. ($8!$ is the total number of ways $8$ people can be ...
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0answers
16 views

Counting permutations of length n without patterns

Count the number of permutations of length n that avoid patterns of high-low-mid. A pattern of hi-lo-mid is 3 integers in the pattern such that for $a_i$,$a_j$,$a_k$, we have i < j < k, $a_i$ > ...
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2answers
34 views

Probability of tossing five coins and getting at least one head

here is my dilemma. I want to know the probability of getting at least one head in five coins being tossed one after the other. Could you help me get the logic of this as it involves both mutually ...
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1answer
47 views

The number of ways of dividing a number by three separate integers.

How many ways can I arrive at the number $45$ by exactly using $5$, $10$ and $20$. I can use each number as many times as necessary. (e.g $9×5$, $20+(5×5)$) this leads to the question, if the number ...
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1answer
31 views

How to interpret combination and permutation problems?

This is more of a methods question than asking for a specific answer: In revisiting statistics and attempting various problems, I am curious if anyone has any insights on how to "see" the route to ...
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1answer
20 views

Multiple Group Representations using Cayley's Thm

I know that an abstract group can be made isomorphic to a subgroup of a symmetric group, by using a Cayley table for that abstract group. However, what is a technique for getting another permutation ...
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2answers
38 views

maths permutation help

An experiment consists of randomly rearranging the 9 letters of the word TARANTULA, where all possible orders of the 9 letters are equally likely. Find the probability of each of the following events: ...