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

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4answers
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How many integer solutions to$ x+y+z \le 6$ where $x,y,z$ are non-negative?

I understand the steps towards working this question out if $x + y + z = 6$, but what are the steps for an inequality?
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1answer
34 views

Number of distinct colourings for the regular pentagon

This is an answer check for the number of distinct colouring's for the regular pentagon given only four colour choices. I have the rotational group action ...
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1answer
39 views

The probability of choosing a particular color combination of balls after five draws

Sorry if this has been asked before, i can't get my head around how to take into account that the order of the balls being drawn is not important.
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1answer
43 views

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 ...
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1answer
47 views

permutation $\pi$, type, permutation $\sigma^4 = \pi$

Permutation $ \pi$ has a signature $2^43^5$. Find number of permutation $\sigma$ such that $\sigma^4 = \pi$ Could you give me a clue ?
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2answers
101 views

Ways to place 7 balls in 14 boxes.

How many ways are there to place 7 balls in 14 boxes. Balls are numbered from 1 to 7. One box can contain only one ball. And out of 14 boxes atleast 6 boxes must contain first 6 balls. 7th ball is ...
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1answer
37 views

Permutation_and_combinations_basic [closed]

In How many ways can we arrange numbers from 1 to 7 in n boxes such that we can repeat any number any number of times?But all the numbers from 1 to 7 must at least appear once. for example - If n = 7 ...
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4answers
47 views

Simple combinatorics dice problem

How would you explain, how ${5\choose4}$ corresponds to the pattern $6-6-6-6-3$ when throwing five dice. The pattern is meant to form a sum of $27$ in total. What I do understand is that you could ...
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1answer
25 views

What is the number of strings satisfying the following constraints?

We need to make a string of size $n$. Each character of the string is either ‘R’, ‘B’ or ‘G’. In the final string there needs to be at least r number of ‘R’, at least b number of ‘B’ and at least g ...
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0answers
45 views

Permutation order statistics integral

Let $U_i$ be $[0,1]$ i.i.d. uniform random variables, for $i=1,\ldots,n$. As an example, let $n=3$. Now pick an ordering, say $x_1>x_2<x_3$. and consider the order statistics integral ...
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1answer
64 views

The number of ways people standing in a line can be holding hands

I'm writing a program to analyze the maximum unique sequences of data in a string, given certain sets of two can be interpreted in two ways. There's a bit of math that I can't figure out, I've ...
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1answer
33 views

In how many ways can 4 balls be distributed in 3 distinct boxes when each box may have any number of balls. Also, 2 balls are identical

I know how to do it for $n$ identical ones being distributed in $k$ distinguishable boxes as $\frac{(n+k-1)!}{n!(k-1)!}$ and for $n$ distinguishable ones being distributed among $k$ boxes as $k^n$, ...
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2answers
55 views

Combinations: People want a beer, there are certain kinds of beer, but limited numbers of each kind

Four people go to a pub and each wants to drink a pint of either the lager, ale, or porter. However, there are only 2 pints of lager, 1 ale, and 1 porter available to drink. How many combinations of ...
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2answers
57 views

Number of ways to put $k$ different numbers in $n$ place

I am working on problem which simplifies to the following which I can't solve. Please help. There are $n$ places and I want to put number $1$ through $k$ in these places. Each place must have one ...
2
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0answers
62 views

Numbers which are writable as a sum of permutation pairs

We say that $N$ is writable as a sum of permutation pair $\{a,b\}$ if $a+b=N$, $a\neq b$ and $a$ and $b$ are permutations of each other (e.g. $321 = 156 + 165 = 147 + 174 = ... $). Looking at 3-digit ...
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3answers
88 views

5-letter strings using the letters in the word “EVERGREEN”

From the word EVERGREEN, 5 letters are chosen at random and arranged into a string of letters. What is the probability that this string is palindromic?
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1answer
52 views

Symmetries on sets of strings

My question is a reference request about symmetries on sets of strings. I'm not a mathematician, so the terminology I use below is probably very non-standard. My apologies. Terminology. Let $[n] = ...
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0answers
43 views

Simple König theorem

I have to prove the "simple" König theorem, without using the marriage theorem: Let $S$ be a set of size $mn$. Suppose that $S$ is partitioned into $m$ subsets, all having size $n$, in two ways: ...
2
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1answer
39 views

differential form identity and permutations

If $t^1,...,t^k$ are the coordinates of a k-cube. Then apparently $$dt^{\sigma(1)} \wedge \ldots \wedge dt^{\sigma(k)}= (\operatorname{sgn} (\sigma)) dt^1 \wedge dt^k $$ I cannot see how this ...
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3answers
47 views

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same.

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same. In other words, putting a book in space 1 and a book in ...
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3answers
78 views

In how many ways can we multiply three numbers? [closed]

Let $a,b,c \in \mathbb{Z^+}$. In how many ways can we multiply this three numbers? At any rate, I'm very grateful for your help!
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3answers
46 views

linear algebra-permutation

Given the permutation $$\sigma = \begin{pmatrix} 1&2&3&4&5\\3&1&2&5&4\end{pmatrix}$$ the matrix A is defined to be the one whose i-th column is the $\sigma(i)$-th ...
3
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1answer
51 views

Bit string combinations [closed]

My madam gave me these questions, can anybody help me? How many bit strings of length $8$ contain: a) exactly three 1s ? b) most three 1s? c) at least three 1s? d) an equal number of 0s and 1s?
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1answer
30 views

Number of ways to divide variables into two categories

I'm looking for a possible solution to find out the maximum number of combinations that can be derived from the given variables. If I'm not mistaken, I think permutations and combinations is the way ...
2
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0answers
50 views

Drawing a Truncated Octahedron

I'm trying to draw a truncated octahedron in MATLAB. This is also known as a permutahedron so my strategy is to link up all the vertices via adjacent transpositions of permutations in $S_4$. What I ...
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1answer
88 views

Maximum and average number of inversions in array by induction

Just for your information, an inversion in an array $a$ is any ordered pair of points $(i, j)$ where $i < j$ and $a_i > a_j$. I can prove the maximum and average number of inversions in an ...
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2answers
40 views

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 ...
0
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1answer
48 views

Fundamental principale of permutations

I have just begin to learn about Permutation and combination. (Just learned definition and factorial.) In which i have learn: $\sideset{_n}{_r}P=n(n-1)(n+1) \dots (n-r+1)$, $r\le n$ where ...
1
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1answer
40 views

Cardinal Arithmetic and a permutation function.

I am working on the following problem and am having difficulties getting started: We define a permutation of $K$ to be any one-to-one function from $K$ onto $K$. We can then define the factorial ...
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1answer
83 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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1answer
57 views

distribution of books among students

There are $p$ students and $q$ books where $q>p$ and all books are different, but each student will get a minimum of $1$ book and a maximum of $(p – 1)$ books. Find the total number of ways of ...
0
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1answer
42 views

Number of ways to choose 6 cards with the same suit from a normal deck of cards

In how many ways can one choose 6 cards from a normal deck of cards so as to have all suits present? One way was $\binom{13}{1}\binom{13}{1}\binom{13}{1}\binom{13}{1}$ but it involves repetition of ...
0
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0answers
31 views

What's the name of $\sum_{k = 0}^{n} (-1)^k {n \choose k} (n-k)^w$?

I worked out the following expression as the number of all possible "words" consisting of exactly $w$ letters from an alphabet $L$ of size $\left|L\right| = n \leq w$, and containing each of these $n$ ...
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1answer
31 views

Count ways to sit men women in row of size K

Suppose we are given N men and M women.They are to sit in a row of size K such that no two women sit next to each other.What are the number of ways. Like if suppose their are 3 men and 2 women and ...
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2answers
74 views

Question of Permutation and combination

I have found a question from somewhere in the internet as follows: English language has 26 alphabets, out of 4 distinct vowels and 7 distinct consonants, how many letter patterns can be made ...
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1answer
48 views

Permutation as a product of transposition

I'm trying to figure out how the proof of the following theorem works: THEOREM: Every permutation is a product of transpositions. The proof is based on noetherian induction. I don't understand how it ...
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1answer
31 views

How Many Unique Character String Can Be Made From 62 Characters?

I'm working on a programming algorithm and need a little math help. I'm in 10th grade and I think the question I'm asking is actually a permutation and combination logic question. Okay, so I've 62 ...
2
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0answers
22 views

'Canonical' form of permutations, product of transpositions

I have such 'canonical' form of permutations: $\prod_{i=0}^n (i \ k_i)$, where $i \leq k_i \leq n$. For example, there are all $6$ permutations of $3$ elements. Of course, some transpositions do ...
6
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1answer
37 views

Equivalent of a sequence in regard to a certain length of a cycle for $\mathfrak{S}_{n}$

Let $n \in \Bbb{N}$ ( for me $0\notin \Bbb{N})$. Find the limit as $n$ tends to $+ \infty$ of the following sequence $$\frac{\alpha_{n}}{n}$$ where $\alpha_{n}$ is the number of permutations of ...
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3answers
56 views

Number of license plates formed by four digits and one letter, qualified.

I need some help with this question: If a license plate for a vehicle consist of five characters: $4$ digits (the first of which cannot be $0$), followed by one letter of the alphabet (which ...
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0answers
55 views

Finding different sum factors of a number

Actual Question : A fair die is thrown k times. What is the probability of sum of k throws to be equal to a number n? My Work: Lets have k buckets, fill-in each bucket with value(1-6) so that the sum ...
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3answers
46 views

What should n be so that the probability is less than 0.5 [duplicate]

n represents the number of people. The probability is that none of these people have a birthday on the same day. Neglect people that are born on 29 February. What should n be so that the ...
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0answers
12 views

How to run a number of tests with differing transitions?

I believe this would be the correct exchange to ask this question. I have a black box with 3 dials, a button, and a display. Each of the dials can be spun to be set to a number on that dial. The ...
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1answer
51 views

Where can I find a set of probability problems?

Is there a database of solved probability problems available? I am currently studying probability (and statistics) and, while I think I have a decent grasp of permutations, combinations, conditional ...
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3answers
134 views

Number of elements of order $2$ in $S_n$

How many elements of order $2$ are there in $S_n$? Using combinatorics I arrived at this: For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} ...
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0answers
364 views

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
2
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0answers
82 views

permutation and combination advanced

I have n sets having values less than 100. I need to find how many arrangements could be made if I pick one element from each set such that in the given arrangement there are no duplicates? NOTE: A ...
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3answers
42 views

Number of $r$ letter words taking letters from a $n$ letter word

I can't figure out how to do questions such as this one, any thoughts? What is the number of four letter words that can be formed from the letters in BUBBLE ...
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0answers
13 views

How to write a function or script uisng python to create the specific permutation on a set of numbers in tuples

(5,6,7),(8,9,10),(11,12,13) the above is the given set of number that define in 3 tuples. The desired permutation should be able to gives result as below. (5,6,7),(8,9,11),(10,12,13) ...
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2answers
95 views

Seemingly simple combinatorial problem

Count all $n$-length strings of digits $0, 1,\dots, m$ that have an equal number of $0$'s and $1$'s. Is there a closed form expression?