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

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2
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1answer
214 views

How many positive integers n can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000?

According to my study guide the answer to the exercise, How many positive integers, (n), can we make with the digits 3, 3, 4, 5, 5, 6, 7, if the number n > 4, 000, 000, : The total of numbers n > ...
0
votes
1answer
60 views

How many different 5 characters words are there with only one letter a?

I just need to clarify my answer to this exercise. This is a permutations exercise. If we define a word to be a string of 5 letters of the English alphabet, regardless of meaning, then mnnnw is a ...
1
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0answers
42 views

Finding a permutation $ \alpha $ given $ \alpha^4 $ [duplicate]

I have the following question: Find a permutation $\alpha ∈ S_7 $ such that $\alpha^4 = (2 1 4 3 5 6 7)$. Is $\alpha$ unique? How should I go about this? I've tried a few different trial and error ...
0
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1answer
59 views

permutations and transpositions in even and odd cases

Say we had some $\sigma = (1, 2)(2, 3)...(n-1 ,n)$ could someone explain why this formula doesn't hold for odd n? For instance, $n = 2m+1$ $\sigma = (1,2)...(2m-2,2m-1)(2m, 2m+1)$, why does that not ...
2
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0answers
41 views

Product of permutations in $S_3$

Consider the elements $(1,2)(2,3)$ and $(2,3)(1,2)$ in $S_3$. We have that they equal $(1,3,2)$ and $(1,2,3)$ respectively. I am unable to make sense out of this. For the first product, let $\sigma = ...
0
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2answers
251 views

Counting four-digit numbers with repeating digits

Of all the four-digit positive integers containing only digits from the set $\{2,4,6,8\}$, what fraction of them have at least one of their digits repeating? Express your answer as a fraction. ...
1
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1answer
30 views

a question concerning subgroup of symmetric group

Suppose $H$ is a transitive subgroup of the symmetric group of $n$ symbols. Show that $n$ divides the order of $H$. I tried to show that some $n$-cycle is in $H$ but this idea did not work.
2
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4answers
691 views

How many of these numbers are divisible by 4?

There is this question that I have no idea where did I make the mistake. Each of the digits 1,1,2,3,3,4,6 is written on a separate card. The seven cards are then laid out in a row to form a 7-digit ...
2
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1answer
79 views

Combination Problem with mulitiple variables

I am new to this, but getting into math more and have a question regarding combinations and permutations with several variables involved. I work for a sales company and this question is based on ...
0
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2answers
97 views

Writing a Permutation as a product of Disjoint Cycles

Write the following as a product of disjoint cycles: $(1 3 2 5 6)(2 3)(4 6 5 1 2)$ I know from my solutions guide that the answer is: $(1 2 4)(3 5)(6)$ but I don't know how to do that. I started ...
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3answers
2k views

A bowl contains 10 red balls and 10 blue balls, A women selects ball at random without looking?

How can we solve this question ? A bowl contains $10$ red balls and $10$ blue balls, and a women picks up balls from the bowl, at random, without looking. A) How many balls must she pickup in ...
0
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0answers
54 views

Using Binomial Theorem to show that (-1)^k C(n,k) [duplicate]

I came across this question on my text book and i didn't know how to prove it. Use the Binomial Theorem to show that $$ 0=\sum_{k=0}^n \ (-1)^ {k} C(n,k)$$
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4answers
411 views

How to tell if two matrices are equal up to a permutation

Given two real rectangular matrices A, B how can I tell if they are equal up to a permutation of their rows/column without trying all possible permutations? (This is closely related to the question I ...
2
votes
3answers
358 views

Give a combinatorial argument to show that C(n,k) = C(n,n-k)

What is combinatorial argument and how can i prove this equation ? As far i understand i think we have to apply the Chu-Vandermonde identity but i am not sure if thats right or not. ...
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2answers
3k views

In how many ways can 20 identical balls be distributed into 4 distinct boxes subject?

I was practicing math exercises on text book and i got stuck in this question ? ...
0
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1answer
42 views

combinatorics :: selecting from variety of groups

in how many ways one or more than one fruit can be selected from 6 varieties of fruits given that there are 5 fruits of each variety? MY TRY : i dont have any clue so i am giving my thoughts MY ...
0
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1answer
141 views

Assignment problem with unique “unemployment” costs

Given $w$ workers and $k < w$ tasks, the usual integer cost matrix $(c_{ij})$ for the cost of assigning worker $i$ to task $j$ and a cost vector $(u_p)$ which assigns any selection $p$ of ...
1
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1answer
26 views

Partioning Mystery

Who has the wisdom to answer the following: 9 distinct marbles distrubted into 4 distinct bags with each bag receiving at least 1 marble,how many ways can this be done? Thankyou for contributing! ...
2
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2answers
75 views

How do I calculate permutations where some values are restricted?

I am curious about the formula for determining the number of combinations there are in a given set where some values are restricted to a certain range. For example, if I have a 10 character, ...
0
votes
0answers
71 views

Summing the product of combinations of matrix elements

I have a situation where I have an $NxN$ matrix $A$ where each element $a_{i,j}\in\mathbb{R}_{\leq 0}$. I would like to consider the set of all collections of elements such that each collection of $N$ ...
1
vote
2answers
532 views

Give the digits $0, 1, 2, 3, 4$, and $5$. How many four digit numbers can be formed if digits can be repeated and contain at least one digit $3$

Given the digits $0, 1, 2, 3, 4$, and $5$. How many four digit numbers can be formed if digits can be repeated and contain at least one digit $3$?
3
votes
1answer
342 views

Distributing $n$ different things among $r$ persons

How can $10$ different pencils be distributed among $3$ students? MY TRY $1$ total ways $= 3^{10}$ MY TRY $2$ $10 \times 9 \times 8 =720$ Which one is correct? If both are wrong what is correct ...
0
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2answers
64 views

Permutation Homework

There are two teams.Two games were played.There are three possible outcomes which are win, lose or draw. how many permutations are there?
2
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1answer
30 views

Permutation query

Would anyone be able to help here with this one ? Let $A = \{a, …, z, A, …, Z, 0, …, 9\}$ be some alphabet and let $$q = q_1, …, q_m \text{ and } w = w_1, …, w_n$$ be finite-length words in $A^*$. ...
1
vote
1answer
30 views

Should I divide this permutation problem into cases or are there any quicker methods?

I have got an idea for the second question but I think my approach is too long and I would like to ask whether there are any other quicker methods? Eight cards are selected with replacement from a ...
0
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2answers
1k views

Outcome possibilities with three teams and three outcomes for each game

So there are six teams (let's say: 1,2,3,4,5,6), and they pair up to face each other, (so three games in total). In each game, one team either wins or their is a tie. Let's set up the teams and their ...
3
votes
2answers
88 views

Find permutation that solves $\;\tau \circ X = \sigma$

I need to find a permutation $X$ that solves $\;\tau \circ X = \sigma,\;$ given $$\tau = \begin{bmatrix} 1 & 2 & 3 & 4 & 5\\ 3 & 4 & 5 & 2 & 1 \end{bmatrix} = ...
0
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2answers
35 views

Partioning/Enumeration

How many ways can one distribute A) 15 Balls into 3 bags. Both bag and balls are distinct (labelled) and each bag must contain at least one ball. B) 10 balls into 3 bags. again both bag and balls ...
0
votes
1answer
38 views

Number of k-permutations that have odd number of an element

I want to find a recurrence relation $h_k$ for the number of k-permutations of $\{\infty a,\infty b, \infty c, \infty d \}$ that have an odd number of a's. I let $h_0=0$ because there is no odd ...
0
votes
3answers
840 views

Combination of Password Question

Suppose that a password for a computer system must have at least 8, but not more than 12, characters, where each character in the password is a lowercase English letter, an uppercase English letter, a ...
0
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1answer
73 views

Alternating Pair

I want to find the number of permutations of $1,2,\ldots,N$ having exactly $k$ triples satisfying the condition that either $n_{i-1}>n_i<n_{i+1}$ or $n_{i-1}<n_i>n_{i+1}.$ For example for ...
0
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2answers
112 views

List of all elements of $A_4$ - Jamie Mulholland p. 85

p. 72: $m$-cycle $\iff m - 1$ transpositions. Hence 3-cycle $\iff 2$ transpositions. I condone all the calculations overhead, but I don't understand the proof blueprint. (1.) How do you ...
1
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1answer
68 views

Finding Required Permutation

I have numbers from $1$..$n$. I want to find number of permutation from all $n!$ permutation where the numbers have following arrangement. $L$ $G$ $L$ $G$ $L$ or $G$ $L$ $G$ $L$ $G$. Where L means ...
0
votes
1answer
59 views

Total number of ways to color a regular graph.

I have problem stating "Find total number of ways to color a regular pentagon with 5 colors." If we consider(Exact 5 colors to color the graph) it unlabeled graph then it will be the same to ...
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2answers
50 views

Fraleigh Section 9 Question 27

Question 27 on Section 9 of Fraleigh 7th edition: Part (a) Asks us to prove that a permutation in $S_n$ can be written as a product of at most $n - 1$ transpositions. I feel that this is not true. ...
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2answers
69 views

Permutation/Combination Question

A three digit number is to be formed by using the digit from 1 to 9 without repetition, find the number of three digit numbers that can be formed if the units digit is an odd number, the hundreds ...
2
votes
1answer
98 views

Expected Value of this function

Let’s consider a random permutation p1, p2, …, pN of numbers 1, 2, …, N and Function F is calculated as F=(X[2]+…+X[N-1])^K, where ...
2
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0answers
117 views

Number of permutations when combining two sets?

I have two sets $\{a_{1},\ldots,a_{K}\}$ and $\{b_{1},\ldots,b_{L}\}$, where I know that $a_{1} \leq a_{2} \leq \cdots \leq a_{K}$ and $b_{1} \leq b_{2} \leq \cdots \leq b_{L}$, but do not know the ...
5
votes
2answers
212 views

Vertex-transitive polytope with large facet

Consider a vertex-transitive convex polytope with a facet containing more than the half of all vertices. Does it already have to be a simplex or are there other examples? I am particularly interested ...
0
votes
1answer
130 views

Number of permutations for n elements with different probabilities

I'm studying the paper Database-friendly random projections: Johnson-Lindenstrauss with binary coins by D. Achlioptas and can't manage to work out the total number of permutations with repetitions in ...
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1answer
334 views

ln how manyways can we distribute $7$ apples and $6$ oranges among $4$ children so that each child gets at least one apple.

In how many ways can we distribute $7$ apples and $6$ oranges among $4$ children so that each child gets at least one apple? I think this can be solved by using permutations because the word ...
1
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0answers
80 views

A company has 20 employees, 12 male and 8 female. How many ways are there to form a 5 person committee?

A company has 20 employees, 12 male and 8 female. How many ways are there to form a 5 person committee that contains at least one male and at least one female? Is this right? no. of ways to select 5 ...
0
votes
2answers
317 views

In how many ways can we arrange 40 boys and 20 girls in 5 groups of 12 members each, so that each group contains at least one girl.

My approach There are 5 groups with 12 members each,so if there was condition like there should be 3 girls and 2 boys i would do (20C3)*(40C2) But here it is given as atleast one girl,how to ...
0
votes
1answer
43 views

K×H is Isomorphic to A4?

Prove \ Disapprove : There exist two non-trivial sub groups $K$ and $H$ such that $A_4\cong K×H$: My intuition was to disprove this claim by saying that $H$ or $K$ must be the Klein sub-group and ...
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2answers
68 views

Abstract Algebra - Permutations

I'm asked to show that $(1,2,3) \in S_3$ generates a subgroup which is normal. I know that I could show it explicitly but that would be tedious. I think it may have to do with the fact that $(1,2,3)$ ...
2
votes
0answers
78 views

How many different ways can 10 octupuses touch legs?

There are 10 octopuses (octopi?). Each octopus has 8 legs. Legs on an octopus can only touch touch legs on other octupuses. Assuming each leg touches exactly 1 other leg, how many different ...
4
votes
2answers
449 views

Expected Value of Local Maxima and Local Minima

Recently I came across this question: Given a random permutation of integers 1, 2, 3, …, n with a discrete, uniform distribution, find the expected number of local maxima. (A number is a local maxima ...
0
votes
1answer
22 views

Combination - Ordering Boy Scouts

In how many different ways can 9 distinct boy scouts be arranged in a 3 × 3 formation? In such a formation, there are 3 scouts in the first row, 3 in the second, and 3 in the third. Two formations are ...
3
votes
4answers
443 views

Permutations of a word with repetitions and conditions

How many permutations of "committee" exist where is must not end in an 'e' ? I've been trying to figure out a possible angle of attack on this question. I've tried to say instead, "how many ...
0
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1answer
87 views

How many permutations of the sequence 1, 2, 3…N where none of the first K numbers in the original sequence is in it's place?

For the sequence 1, 2, 3 ... N there are of-course N! permutations. But for a given K, where 1 < K ≤ N how many permutations are there given none of ...