# Questions tagged [permutations]

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

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### Algorithm for Gaussian elimination to permutation matrix over $\mathbb{Z}_2$

I wish to find an algorithm that reduces a matrix over $\mathbb{Z}_2$ to a permutation matrix using as few row summing operations as possible. Standard Gaussian elimination turns a matrix into the ...
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### A question about weights

I have a question, but I'm not sure if this question is for this group or another ( I seem to remember there's another group for this kind of questions though ): A guy has $100$ stones, and they're ...
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### In how many ways can 20 persons be seated round a table if there are 9 chairs?

The problem given is in the title: In how many ways can $20$ persons be seated round a table if there are $9$ chairs? I tried solving it as follows: I can fix one person to one of $9$ chairs. I ...
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### Decomposing a cycle into transpositions

I am trying to go through some lecture notes found here, and am struggling with an assertion on page 21. It is meant to be a proof that any cycle $(a_1,..,a_k)$ can be written as a product of ...
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### Among the numbers 1,2,3,4,5 and 6 how many 4 number combinations can be made so that there would not be three consecutive numbers in a row? [on hold]

Probability theory and need to find the combinations and it's the answer
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### number of ways of selecting n balls from m buckets with each having different number of balls [duplicate]

Imagine you have been given n different colored balls and a task to choose at most k balls from them such that none of the same colored balls is selected for any given sequence and considering same ...
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### Among the numerals 1., 2, 3, 4, 5, and 6 how many 4- number combinations can be made so that there would not be three consecutive numbers in a row? [on hold]

Among the numerals 1, 2, 3, 4, 5, and 6 how many 4- number combinations can be made so that there would not be three consecutive numbers in a row?
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### How many ways can you arrange 3 pennies and 2 nickels in a line?

All coins of the same type are indistinguishable. So I start with a 5 choose 3 for the pennies. 5!/(3!2!). Giving me 10 ways to arrange the pennies. For each combination of the pennies it ...
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### Worst case for algorithm solving a game

I came up with the following algorithm on a whim, and it doesn't relate to any deeper mathematics, but I'd like to see if someone can prove a connection it has to a sequence I found. You can think of ...
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### Finding matrix representation of linear transformation with respect to permuted basis

I'm struggling with understanding how I can figure out exactly what the following matrix representation looks like: Let $V$ be an $n$-dimensional vector space. Let $\mathcal{B} = (b_1,\ldots,b_n)$ ...
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### Multiplication of cyclic permutation

Find $f^{-1}gf$, where $f=(123)$, $g=(2345)$. Since $f$ and $g$ are cyclic permutations, we have $gf=fg$. Hence $f^{-1}gf=f^{-1}fg= (f^{-1}f)g=ig=g=(2345)$, where $i$ is identity permutation. Since i ...
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### How many n-bit binary numbers can be formed such that, for every prefix (position) the number of 0's is at least 'x' times than the number of 1's?

For ex- (1) if n = 5 and x=3 Then 3 such binary nos. are possible, which are- 00000, 00001, 00010 All these nos have no of 0's at least 3 times the no of 1's at any prfix(position) (2) if n=6 and x=...
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### The best $n$-digit password?

I suddenly thought of a question today: What is the best $n$-digit password? It is not specific so I'll write it in a better way: There is a password lock that has $n$ digits. There are $t$ choices ...
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### Find number of ways to select subset with distinct objects of at most K size.

I have a set of x number of 'A' type object, y number of 'B' type object and z number of 'C' type object. Now, I need to find the number of subsets of atmost 'k' size that can be made such that all ...
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### Proving a matrix has Kronecker product form

Is there a property unique to Kronecker product of two matrices, so that one could use it to prove that a certain matrix has Kronecker product form? Is there a proof technique to this type of ...
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### Permutations question - very confused and our teacher has taught us nothing. Any input would be helpful so thanks very much! [closed]

A computer randomly selects 4 digits from between 2 and 8 (includes both 2 and 8) a) repetitions are not allowed b) no repetitions are allowed and it must start with an 8? c) no repetitions are ...
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### Permutations Questions - How many ways to order books with constraint?

Question: There are 50 books. Dan tries to order them. His only constraint is $5$ books by Dan Brown that need to be next to each other. How many possible orders for the $50$ books are there ...
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### Permutations on circular table so that i cannot go to i or i+1.

Suppose $n$ objects are placed in a circular table in clockwise order. Find the no of permutations where $i$ cannot go to $i$ or $i+1$. i.e. $1$ cannot be mapped to $1$ or $2$, $2$ cannot be mapped to ...
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### Over-Counted Permutations

Setup We have 4 unique cards I am trying to find the probability that none of them are in the same position after shuffling The Given Solution My Questions They reason that there are 4 ...
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### How to interpret this combination formula

If you have $m$ and $n$ number of distinct items, and you get to choose $p$ and $q$ things respectively out of them, then the number of ways you can permute $p+q$ items is as follows: $m$ items can ...
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### Number of arrangements of one $1$, two $2$'s, three $3$'s, four $4$'s, …, and nine $9$'s (given constraints)

We have no zeros, one $1$, two $2$'s, three $3$'s, four $4$'s, five $5$'s, six $6$'s, seven $7$'s, eight $8$'s, and nine $9$'s. How many $12$-digit numbers can we make using at least three $6$'s so ...
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### In how many ways can $14$ people be seated in a row if there are $8$ men and they must sit next to one another?

In how many ways can 14 people be seated in a row if: a.) there are 7 men and 7 women and no two men or two women sit next to each other? My attempt: Since no two men or women can sit next to each ...
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### Count Subsets of size less than equal to k [duplicate]

This is a variation of question asked on this site before. Consider a set with $𝑎_1$ 'distinct' 1s, $𝑎_2$ 'distinct' 2s, ... , $𝑎_𝑛$ 'distinct' ns. You have $𝑎_1+1$ choices for the 1s (including ...
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### Number of three-digit even numbers with no repeat condition.

I have to find the total number of three-digit even numbers where no digit can be repeated. I tried and got answer $9 \times 9 \times 5$, but it is wrong. There is something weird with $2$ digits. I ...
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### What is the connection between number of permutations and number of subsets? [closed]

How many different ways to fill 100 boxes in a line with black or white balls. (One box can only contain one ball at a time.) My attempt : Different ways to fill 1 st box = 2 Different ways to fill ...
### How many ways can $2$ different history books, $5$ different math books, and $4$ different novels be arranged on a shelf if …?
This is my first class in probability so I just wanted verification as to my attempted solution. Question: In how many ways can $2$ different history books, $5$ different math books, and $4$ ...