Tagged Questions
1
vote
1answer
24 views
possible combinations of 3-digit
How many possible combinations can a 3-digit safe code have?
Because there are 10 digits and we have to choice 3 digits from this,
then we may get $10^P3$ but A author used the formula $n^r$, why is ...
0
votes
3answers
30 views
permutation/combination problem
There are 3 doors to a lecture room. In how many ways can a lecturer enter the room from one door and leave from another door?
I have done like this: They way of entering is 3 and exiting is also
...
5
votes
1answer
58 views
Is there a name for this given type of matrix?
Given a finite set of symbols, say $\Omega=\{1,\ldots,n\}$, is there a name for an $n\times m$ matrix $A$ such that every column of $A$ contains each elements of $\Omega$?
(The motivation for this ...
6
votes
1answer
40 views
Grouping natural numbers into arithmetic progression
I need to find the number of ways of dividing the first 12 natural numbers into 3 equal groups (4 numbers each), so that the numbers in any particular group can be arranged in AP (Arithmetic ...
2
votes
1answer
22 views
Number of Total orders of a dependency graph
Define a dependency graph to be a graph $G=(V,E)$ such that an edge between vertices $v$ and $u$ in $V$ is present if $v<u$ i.e. $v$ comes before $u$ in our ordering (I'm not very concise here, I ...
1
vote
2answers
66 views
Permutations of a queue of interlaced boys and girls.
Suppose $5$ boys and $4$ girls are to be arranged in a queue such that between any
two boys there is at least one girl. Find the number of such arrangements possible.
What i think is $5$ boys ...
2
votes
6answers
51 views
calculate the number of possible number of words
If one word can be at most 63 characters long. It can be combination of :
letters from a to z
numbers from 0 to 9
hyphen - but only if not in the first or the last character of the word
I'm trying ...
0
votes
0answers
57 views
Ball and holder problem [duplicate]
I am trying to solve this but having a tough time deriving the formula.
There are $X$ ball and $Y$ holders $Y \leq X$. Out of the $X$ balls, $N$ are red and $X-N$ are blue.
What is the probability ...
1
vote
1answer
30 views
Need an algorithm to compute number of elements in sample space
An urn contains $X$ red balls, $Y$ green balls, and $Z$ white balls. $N$ balls
are drawn without replacement from the urn, and the colors are noted in sequence.
$N \leq X+Y+Z$
Trying to come up ...
1
vote
1answer
23 views
Dice Roll Permutation Problem
Here is my problem:
You have a standard dice, with possible rolls: $\{1, 2, 3, 4, 5, 6\}$. How many permutations exist in 10 rolls such that no two immediate rolls are the same?
For example:
$\{1, ...
2
votes
2answers
35 views
From Combination to Permutation
I am facing a (probably) basic counting issue.
If $P(n,r)$ the permutations for $r$ objects from $n$ and $C(n,r)$ the combinations, we have : $P(n,r) = r!C(n,r)$.
Yet there are two example in which ...
2
votes
1answer
45 views
Permutation & Combination - How many 4 digit no's are there whose decimal notation contains not more than two distinct digits?
How many 4 digit no's are there whose decimal notation contains not more than two distinct digits?
1
vote
3answers
62 views
The number of bijections $f$ of $\{1, 2,…, n\}$ such that $f(i) \ne i$ for any $i$
Show that the number of bijections $f$ of $\{1, 2,..., n\}$ such that
$f(i) \ne i$ for any $i$ is equal to $$\sum_{j=0}^{n}(-1)^j\frac{n!}{j!}.$$
Can I get some help for the above problem? I am ...
1
vote
1answer
27 views
On permutations and Combinations
$mn$ squares of equal size are arranged to forma a rectangle of dimension $m$ by $n$, where $m$ and $n$ are natural numbers.
Two squares will be called 'neighbours' if they have exactly one common ...
3
votes
2answers
54 views
Unable to get to all permutations after $n-1$ transpositions
Problem: Give an example of a permutation of the first $n$ natural numbers from which it is impossible to get to the standard permutation $1,2,\ldots,n$ after less than $n-1$ transposition operations ...
2
votes
2answers
45 views
Number of rectangles with odd side lengths on a chess board?
Given an 8x8 chess board, how do we find the total number of rectangles with odd side lengths?
(Both sides have odd length).
In general, what would be an elegant method to deal with problems like ...
0
votes
1answer
42 views
Counting Methods: Restricted Permutations
I have been scratching my head for a long time. The question is: How many words can be formed using all letters in the word EXAMINATION in such a way that the first two letters are different ...
0
votes
2answers
49 views
How many ways are there to encode the 26-letter English Alphabet into 8-bit binary words?
I know that I need 5 bits to represent a character. All the combinations to encode the 26-letter alphabet will be 2^5? How about the 3 bits that remains from 8 bits?
1
vote
0answers
24 views
Notation for Restriction of Permutation
Suppose $\sigma$ and $\tau$ are permutations such that $\sigma(x)\not=x\implies \sigma(x)=\tau(x)$. Intuitively, I would like to think of $\sigma$ as a restriction (or projection) of $\tau$ onto a ...
0
votes
1answer
13 views
finding out team number
A supervisor has to select a three-member project team from among her 12 employees. Unfortunately, two of the employees cannot work together on the same team. With this restriction, how many different ...
0
votes
2answers
34 views
Determine no of combinations for cutting stock algorithm
I have to buy $n$ wooden logs of size 2000 each, from which I have to cut different pieces of smaller size say:
255*10
750*7
550*13
In a manner that cutting will ...
4
votes
3answers
85 views
Distributing identical objects to identical boxes
We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?
4
votes
1answer
65 views
Find a lower bound
Let $M$ be an $N\times N$ symmetric real matrix, and let $J$ be a permutation of the integers from 1 to $N$, with the following properties:
$J:\{1,...,N\}\rightarrow\{1,...,N\}$ is one-to-one.
$J$ ...
3
votes
1answer
47 views
Is there a name for this type of permutation?
Let $J$ be a permutation of the first $N$ integers (1, 2, ..., $N$), so that the permuted sequence reads $(J(1),J(2),...,J(N))$. The function $J$ must of course be a bijection. Additionally, suppose ...
3
votes
2answers
32 views
Number of Strings with two specific letters
How many ways can you construct a string four letters (from 26 alphabet characters) that have both the letters j and k in them?
2
votes
1answer
51 views
Permutation Formula
I am having difficulty with one minuscule detail of the permutation formula:
$$n(n-1)(n-2)\cdots(n-r+1)$$
I understand that if we proceed with an $r$-permutation, then we have $r$ amount of slots, ...
3
votes
3answers
38 views
Discrete Math, anagram combinatorics
Find the number of anagrams for the word "ALIVE" so that the letter "A" is before the letter "E" or the letter "E" is before the letter "I". By before we mean any letter previous, not just immediately ...
1
vote
2answers
43 views
Inclusion & Exclusion: In how many permutations of the digits $0,…,9$ there's no continuity of 7 digits or more?
In how many permutations of the digits $0,...,9$ there's no continuity of 7 digits or more?
(Ex. the number 203456789 1 should not be counted)
I believe that the basic case, for the inclusion ...
7
votes
1answer
89 views
How do combinations (not permutations) relate to group theory?
First question. I'm just generally curious about combinations in group theory. How do they relate?
If I take the set of permutations of $\langle 1,2,3,4 \rangle$, I get the symmetry group S4. How ...
0
votes
0answers
35 views
Placing red and black balls in n places
There are n places and there are c1 number of black balls with each numbered from 1 to c1 and ...
0
votes
1answer
47 views
number of ways of placing balls on plate
There are n plates places in a line and unlimited number of red balls with values from 1 to ...
1
vote
1answer
55 views
Doubt on married couple seating arrangement problem
I am going through a solution of the following problem. "How many ways there are there to seat $n$ couples around a circular table with $2n$ chairs such that no couple sits next to each other, i.e., ...
1
vote
1answer
26 views
Number of arrangements around a table
My doubt is based on two observations :
1) On top of a round table (which is rotatable) there are $n$ places to sit and we need to place $n$ people. How many ways is it possible to permute them ?
...
1
vote
2answers
22 views
mixed permutations and combinations
I have a problem that I am not too sure of. In a team of 16, there are 5 couples and 6 single people. In how many ways can at most 1 couple be chosen if 6 people are required to represent the team at ...
0
votes
3answers
132 views
How many bit strings of length 17 contain at least 5 ones?
I'm a really confused as to how to start this question, would really appreciate any help you guys could give me!
5
votes
1answer
114 views
Why is $S_5$ generated by any combination of a transposition and a 5-cycle?
Why is $S_5$ generated by any combination of a transposition and a 5-cycle? Is this true for any prime $p$ (in this case $p=5$)?
-4
votes
1answer
38 views
How many strings(letters and digits) of length 6 have exactly one digit?
How will I solve this? Is it possible to use permutaions as well?
3
votes
0answers
45 views
The Mathematics of Shuffling Poker Chips?
First, I must say that I do not have an advanced understanding of mathematics and I don't know what category this question belongs in. This is just a question that I have been thinking about recently.
...
2
votes
1answer
70 views
Permutation & Combination
There is a game in which there is a point P and k other points on a plane. To win, we must draw directed lines starting from point P and ending at point P with exactly n number of lines to be drawn. P ...
1
vote
1answer
80 views
Exponential generating function for permutations with descent set whose least element is even
Let $E(n)$ be the number of permutations $w\in S_n$ such that the least element of the set $Des(w)\cup \{n\}$ is even, where $Des(w)$ is the descent set of $w$. I need to find the exponential ...
2
votes
3answers
41 views
Computing $\langle (13746) \rangle$ in $S_7$.
How to list the elements of subgroup $\langle a \rangle$ in $S^7$ where $$a=\begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 3 & 2 & 7 & 6 & 5 & 1 &4 ...
1
vote
3answers
63 views
Restricted Permutations
I have a problem in which 4 music books, 5 education books and 2 medicine books need to be arranged on a shelf. In how many ways can this be done if only the music books must be kept together and all ...
1
vote
2answers
140 views
Probability of having $k$ empty urns after putting $n$ balls into $n$ urns
Assume that there are $n$ balls (numbered from $1$ to $n$) and $n$ urns (numbered from $1$ to $n$). At the beginning no ball is placed in any urn. Balls are randomly thrown into urns: Each ball is ...
0
votes
1answer
36 views
On Circular Permutations
In how many ways can 3 ladies and 3 gents be seated together at a round table so that any two and only two of the ladies sit together?
1
vote
1answer
60 views
Restricted Permutations and Combinations
Tino, Colin, Candice, Derek, Esther, Mary and Ronald are famous artist. Starting next week, they will take turns to display their work
and each artist's work will be on display at the London Show for
...
4
votes
2answers
52 views
How to determine the number of 5 consecutive digit blocks in a set of digits?
Let there be a set containing the following digits: {1,2,3,4,5,6,7,8,9}.
If I choose 5 digit blocks, where the digits are arrange in consecutive ascending order, ...
3
votes
2answers
76 views
Derangements property
It is not difficult to evaluate a formula for the number of derangements, with a simple combinatorical argument we get $D(n)=(n-1)(D(n-1)+D(n-2)), n\ge 3$ where $D(n)$ is the number of derangements.
...
9
votes
2answers
139 views
Cocktail bar problem
Recently I went out with friends and we asked ourselves the following question: Consider $n$ people sitting at a cocktail bar next to each other. How many rearrangements have to be made to ensure that ...
7
votes
0answers
139 views
Citation for subset complement result
Let $S = \{s_1, \ldots, s_n\} \subset \{1, \ldots, 2n\}$. Consider two operations on $S$, unfortunately both called complement in different setting: let $A(S) = \{1, \ldots, 2n\} \setminus S$ (set ...
1
vote
2answers
36 views
Permutations in Two Rows
I have been looking at linear and circular permutations. I have now come across a question that entails permutations in two rows.
This is the question:
Six natives and two foreigners are seated in a ...
