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

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combination and permutation with restraint

There is a group of 10 objects, 2 red, 3 blue and 5 green. If the red should be one at the beginning of the line and the other at the end, calculate how many combinations. As the two reds are ...
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1answer
24 views

Combination and permutation of indistinguishable objects

There is a group of 10 objects, 2 red, 3 blue and 5 green. The objects are indistinguishable. In how many ways can they be arranged on a line? As there are 3 groups of objects I did that: $ 10! / ...
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2answers
33 views

52-Deck 3 Cards Drawn Possible Combinations Question

I have a HW problem I'm trying to pin down and I think I'm confusing myself... Question: In a card game w/ a standard 52 card deck, a hand is a set of 3 cards. Count the # of hands that are... a) ...
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3answers
28 views

Probability Urn Group Problem

my group and I are having trouble figuring out how to do this. For some reason I have an urn that contains 10 coins. 3 of the coins are blue on one side and red on the other, 3 of the coins are blue ...
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1answer
78 views

$n!$ as a sum of $n$ positive integers

We partition $(n-1)!$ into $n-1$ parts in the following way. Consider a permutation $(a_1,a_2,\ldots,a_n)$ of $(1,2,\ldots,n)$. We say that $a_k$ dominates its predecessors if $a_j<a_k$ for ...
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1answer
25 views

What permutations of matrix entries do row and column transpositions generate?

Let $M$ be a square matrix. By transposing rows and columns, can we get any permutation of the entries of $M?$ If we can't, which permutations are generated?
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0answers
35 views

The number of bijective polynomials of particular degree in a field

I need to know please: In a finite field of q elements how many bijective polynomials exist whose degree are smaller than d?
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4answers
81 views

Number of length $8$ binary strings with no consecutive $0$'s

How many $8$ bit strings are there with no consecutive $0$'s? I just sat an exam, and the only question I think I got wrong was the above(The decider for a high-distinction or a distinction :SSS) I ...
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1answer
27 views

How can i learn when to use which multiplication rule: Probability

Hey guys im studying for a math exam and was wondering if anyone has some easy techniques to remember in what kind of scenario to use these equations. These are I believe called multiplication rules. ...
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2answers
24 views

Permutation question on alphabets

Ten different letters of alphabet are given, words with 5 letters are formed from these given letters. Then, the number of words which have at least one letter repeated is Well I do understand some ...
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4answers
37 views

Arithmetic of a combinations formula

I am trying to study, and I'm not quite sure how: $$ \binom{5}{3} \cdot \binom{7}{3} = 350 $$ From my understanding the formula is $$ \binom{n}{r} = \frac{n!}{r!(n-r)!} $$ Therefore: $$ ...
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0answers
24 views

Number of permutations if every element may appear in certain distance from its initial position

Suppose i have an $n-$elements array. I want to count number of permutations for which element $a_i$ allowed to appear in range $i-k, \dots, i+k,$ so $2k+1$ positions available after permutation has ...
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1answer
15 views

Difference between non - negative and positive integral solution :

Difference between non - negative and positive integral solution : (a) Number of non negative integral solution of equation $x+2y+3z+4w =n$ = Coefficient of $x^n$ in ...
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1answer
29 views

How many permutations of the letters in HURRAH have the first R preceding the first H?

How many permutations of the letters in HURRAH have the first R preceding the first H? This is equivalent to the number of permutations with $R$ in the first position + The number of permutations ...
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0answers
57 views

Finding the initial permutation

Assume a permutation P[1], P[2], ..., P[n]. Now we have made N permutations Q[1], Q[2], ..., Q[N], Q[i] is permutation P without element i (we subtract 1 from all elements bigger than i). For ...
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1answer
31 views

Arrangements of the word HULLABALOO

Three problems below with my attempt solutions: 1) How many ways (ordered selections) can the letters of the word HULLABALOO be arranged? $$\frac{10!}{3!2!2!}$$ 2) How many distinguishable ...
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1answer
20 views

Shuffling cards alternately return to original position

There are $2n$ cards in a stack. At any stage, if the cards are $a_1,a_2,\ldots,a_{2n}$ from top to bottom, then it becomes $a_{2n},a_1,a_{2n-1},a_2,\ldots,a_{n+1},a_n$. In how many steps will the ...
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1answer
32 views

Counting the number of permutations where $i$ does not follow $i-1$

Whilst reviewing, I've found a problem where the book has an answer, but no suitable explanation, and I can't begin to connect the two. The problem is simply as follows: Count the number of ...
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1answer
20 views

Product of differences of permutation in circle

Numbers $1,2,\ldots,n$ are arranged into a circle. What is the maximum product of the differences $|x_1-x_2|\times|x_2-x_3|\times\cdots\times|x_{n-1}-x_n|\times|x_n-x_1|$? I think the maximum should ...
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2answers
35 views

a question about abstract algebra, a question related to permutation

Given $X=\{1,2,......n\}$, let us call a permutation $p$ of $X$ an adjacency if it is a transposition of the form $(i,i+1)$ for $i\le n-1$. Prove that $(i,j)$ is a product of an odd number of ...
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1answer
41 views

Sum of differences of permutation in circle

Numbers $1,2,\ldots,n$ are arranged into a circle. What is the maximum sum of the differences $|x_1-x_2|+|x_2-x_3|+\ldots+|x_{n-1}-x_n|+|x_n-x_1|$? I think the maximum should occur when the numbers ...
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3answers
57 views

Recovering the original values from given information.

We have some N numbers[1..N] and N students. Originally we assign each number to single student. Call this assignment as the initial state of the assignments. Instance of assignment is described as ...
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1answer
34 views

Identifying this kind of combinatoric permutation

I feel kind of silly asking this, but I am having a hard time identifying what this exactly called. I'm specifically trying to find the wiki page on it and could not find it on this list of ...
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2answers
41 views

Count how many numbers containing number 4 among the numbers 1 to 8900?

I Got a Problem like this "Count how many numbers containing number 4 among the numbers 1 to 8900 ?" what is the appropriate formula to solve the problem above, I tried to solve it by creating a ...
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0answers
29 views

The number of Permutation polynomial in a field

I need to know,please: (1) How many permutation polynomial exist in a finite field (any field)? (2) Is there any way to pick a random permutation polynomial in this field?
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1answer
15 views

How many different combinations are possible

Say there are 4 stations for ice cream toppings. station1: 3 choices station2: 2 choices station3: 2 choices station4: 3 choices How many different combinations ...
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1answer
21 views

Permutations and Combinations [closed]

How many ways are there to distribute 9 different video games to 3 people (John, Harry and Fred) with each of them receiving odd numbers?
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2answers
14 views

count number of permutation that map no even number to itself

Problem: How many permutations of numbers $1, 2, ..., 10$ exist that map no even number to itself? I understand that this is "Hatcheck lady" problem. But I am a bit confused how to solve it. So my ...
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0answers
19 views

How to to minimize a sum by changing summation order

I have two vectors $(x_1,\dots,x_n),(y_1,\dots,y_n) \in \mathbb{R}^{n}$. I want to find a permutation $\sigma$ such that $$ \sum_{i=1}^n |x_i -y_{\sigma(i)}|^2$$ is minimized. Is there a better way ...
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1answer
36 views

In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. …

Problem : In an examination the maximum marks for each of the three papers are 50 each. Maximum marks for the fourth paper are 100. Find the number of ways in which the candidate can score 60 % marks ...
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1answer
35 views

Binomial Coefficient Combinations

I have tried to figure this out and I cannot. The professor gave us an answer of 13,536 but I do not see any way in which he got to his answer. Any help would be greatly appreciated. A certain ...
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1answer
21 views

If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the …

Problem : If r,s,t are prime numbers and p,q are the positive integers such that LCM of p,q is $r^2s^4t^2$ then find the number of ordered pairs (p,q)? Can we use this : let $r^2s^4t^2$ = ...
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1answer
41 views

Exponential Generating Function for Permutation with no Fixed Points

While reviewing, I've come across a problem that seems to outline my lack of knowledge with regards to (specifically exponential) generating functions. For some reason, I understand "ordinary" ...
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2answers
22 views
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1answer
19 views

Find the number of ways in which 10 different books can be shared between a boy and a girl if each is to receive an even number of books.

The right answer is 510 but my calculations keep giving me 252. where did I go wrong? please be thorough because I really do not understand this topic.
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0answers
14 views

List of all permutations/combinations

This is more of a practical than a theorical question, i thought you guys can help me with this So i have these 5 Letters , already in the appropriate order - The only element thats missing is ...
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2answers
31 views

Combinatorics For $4$ Pool Balls

There lie $4$ pool balls on a pool table: two striped and two plain. Two of the pool balls are selected at the same time, at random. Given that one of the selected balls is striped, what's the ...
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1answer
34 views

How many ternary strings of length 4 have exactly one 1?

Answer: Ternary strings have symbols 0, 1, and 2. If there is exactly one 1, then there are 3 positions the one can be in and 2*2*2 ways to fill the other 3 blanks with a 0 or a 2. So the answer is ...
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1answer
23 views

Circular array with minimum absolute difference among adjacent elements

Given a circular array, rearrange the array so that the maximum absolute difference between adjacent elements among all elements is minimum. Can anyone help me with this?
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1answer
31 views

“Stars and Bars” method when all variables are multiple of same number

Find the number of non negative integral solutions of the equation $x + y + z + u = 100$, where $x, y, z$ and $u$ are multiple of $5.$ My approach using stars and bars: Since all are multiples of ...
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1answer
50 views

A Proof Question

Prove : $$\sum_{k=1}^nkp(n,k)=n!\;,$$ where $p(n,k)$ is the number of permutations of $\{1,2,\ldots, n\}$ which have exactly $k$ fixed points. I was using $$p(n,k) = \frac{n!}{(n-p)!}$$ and ...
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4answers
142 views

Problem on selecting group of card from a well shuffled pack of card

I have a problem I'm working on: The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 card to guarantee that three cards are from some same suit is which amount? I got ...
2
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1answer
41 views

Show that the permutation [n, n-1,…, 2,1] has n(n-1) inversions

Show that the permutation $[n, n-1,..., 2,1]$ has $n(n-1)$ inversions How do I show that this is true? Why isn't $(n(n-1))/2$
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2answers
30 views

Find the number of 3 letter words that can be formed from the word 'SERIES'.

To find the number of three letter words that can be formed from the word 'SERIES', with or without meaning and without repetition. The number of permutations if all letters were distinct = ...
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1answer
31 views

Permutation of $n$ women and $m$ men, in a line, where the women dont get along with each other

So the $n$ women can't sit next to each other. So in a straight line how many ways can they be seated? I know this problem is partitioning distinct balls in $n+1$ partitions, out of which $n-1$ of ...
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5answers
730 views

Even Number cards?

There are $15$ cards on a table, marked with an integer $1$ from to $15$ . How many ways can I take cards such that the sum of the numbers on the cards is even? Please help me?
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1answer
31 views

how to calculate these intersections without having to count all combinations

We have the following sets: $X= {(a,b,c,d) ∈S: b< c < d},$ $Y= {(a,b,c,d) ∈S: a< c < d},$ $Z= {(a,b,c,d) ∈S: a< b < d},$ $F= {(a,b,c,d) ∈S: a< b < c},$ Where each of ...
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0answers
14 views

Amount of inversions in permutations.

Let $I_n(k)$ be the number of permutations of $n$ values that have exactly $k$ inversions. The true is expression: $$I_n(k) = I_n\left( \binom n 2 - k\right) $$ but I don't understand why. Please help ...
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3answers
486 views

Math puzzle: 10 digit strings generations

There was a question in a math competition that I attended last year. At the end of competition, I realized that my answer was wrong for the question below and I have never been able to figure out how ...
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1answer
36 views

Number of ways of choosing identical balls

Suppose we are given a bag of $n$ identical red balls, what is the number of ways of choosing $3$ red balls from the bag? I know the answer is $$ \binom n3 $$ but isn't there just one way of choosing ...