Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

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For which $n$ we can divide set $M= \{1,2,3,…,3n\}$ in to $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$?

For which $n$ we can divide set $M= \{1,2,3,...,3n\}$ in to $n$ subsets each with $3$ elements such that in each subset $\{x,y,z\}$ we have $x+y=3z$? Since $x_i+y_i=3z_i$ for each subset $A_i=\{x_i,...
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1answer
41 views

Number of ways in which exactly $1$ person is alive

The following question was given by my teacher in an assignment. There is a group of $6$ persons. The $6$ persons shoot each other, no one shoots himself. Everyone shoots exactly once and no shot is ...
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4answers
32 views

probability of 4 couples sitting opposite to each other at a round table with 8 seats

After all are seated, I though I can look at four of them. There is a $1/8$ probability of the first person sitting in front of its mate, then $1/7$ for the second, $1/6$ for the third and $1/5$ for ...
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2answers
21 views

Question about possibilities of a team

consider an experiment that consists of determining the type of job - either blue-collar or white collar- and the political affiliation -republicans, democratic or independent - of the 15 members of ...
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1answer
27 views

Iberoamerican Olympiad 2005: Determine the number of ways of coloring these $2n$ points

Let $n$ be a fixed positive integer. The points $A_1$, $A_2$, $\ldots$, $A_{2n}$ are on a straight line. Color each point blue or red according to the following procedure: draw $n$ pairwise disjoint ...
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3answers
40 views

How can we count the number of combinations without casework in this problem?

This is an interesting problem that I remembered today: If a billboard can be painted either red, orange, or yellow, and it is never painted red for 3 days in a row, then how many paint-sequences ...
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2answers
59 views

Combinations: How many possible codes does she have

A spy is trying to open a security door by entering the correct code into a key pad. The key pad has 10 buttons (for the digits $0,\dots ,9)$. How many possible codes does she have to try at most if......
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2answers
55 views

A probably simple combinatorics question

I've faced the following problem, it comes from physics actually, but the precise details are of no importance. Consider a set $\{x_{n}\}_{n=1, ..., N}$. We construct the following function $$F(\{x_{n}...
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3answers
69 views

Why is this problem solved using 'stars and bars' theorem?

We have a set of $n^2$ integer and each number of them is in interval $[1, n]$. Every number from $1$ to $n$ is frequent $n$ times. For example, $n = 3$ and set is $\{1, 1, 1, 2, 2, 2, 3, 3, 3\}$. We ...
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25 views

Partitioning a rectangle or tiling it with rectangles

Can we count or even recursively construct all ways to tile or partition a rectangle with rectangles? Many other questions ask either of two things none of which I am after: Partitioning a rectangle ...
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26 views

Derangements of the letters of the word PURPLE [duplicate]

In a Youtube video on the channel @blackpenredpen, the number of derangements of the word "PURPLE" is calculated as $$\frac{!6 \,- \,!5\, -\, !5\, - \,!4}{2!}$$ I don't understand two ...
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17 views

Minimal number of intersections of families of lines

I'm considering two families, $F_1$ and $F_2$, of lines in the plane with $\vert F_1 \vert= N_1$ and $\vert F_2 \vert =N_2$. The families are such that if we pick $g \in F_1$ and $l \in F_2$ we get ...
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0answers
20 views

Given a necklace with length $N$, how many ways to color it using 2 colors, such that any rotation should be counted only once? [duplicate]

My problem is this: Given a necklace with length $N$, how many ways to color it using $2$ colors, such that any rotation should be counted only once? For example, following $2$ necklaces should only ...
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22 views

Is it unordered selection without replacment?

Following my question here, I tried to review my statics course which I took over a decade ago. (sorry forgot most of it, I rarely use it) In any case the new conditions are as follows: 47 apartments ...
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1answer
35 views

Minimum for combinatorial sortingproblem

I'm stuck with a combinatorial problem, maybe one of you can help me out, thanks in advance. So heres the problem: Consider tuples $(i,j)\in \{1,...,N_1\}\times\{1,....,N_2\}=A$. Let $S_1,...,S_x$ be ...
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1answer
11 views

Inner product of a doubly stochastic matrix by a non-negative matrix

Let $W = (w_{ij})$ be a $n \times n$ non-negative matrix and define a function $f$ in the set of all doubly stochastic matrices $n \times n$ by setting for $A = (a_{ij})$, $$f(A) = \sum_{i,j} a_{ij}w_{...
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0answers
33 views

Number of $n$ digit integers where digits appear an odd number of times

Another question asked about the convergence of the sum of reciprocals of integers where digits appear only an odd number of times. (Digits which do not appear are ignored, so for example $14404$ is ...
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0answers
21 views

Knitting pattern with five different colors.

I have 5 different colors: Red, green, blue, yellow, purple. I am going to use 3 colors at once, then change one of the colors, then after a while change one more of the 3 colors. This pattern ...
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3answers
67 views

Amount of numbers $n \in \{1,\ldots,9999\}$ such that there aren't $2$ consecutive odd digits

I want to find out the amount of numbers $n \in \{1,\ldots,9999\}$ such that there aren't $2$ consecutive odd digits. I want to use the principle of inclusion and exclusion. There are 9999 numbers in $...
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1answer
28 views

How many license plate numbers contain the digit 4 exactly once in the string of $3$ digits (after the region code)?

So I have been stuck on this question for a few days now and I am honestly just floored at how to get the correct answer. The question is as follows: A license plate number consists of eight ...
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0answers
44 views

The congruent mod 5 relation partitions the integers into following five classes :

The congruent mod 5 relation partitions the integers into following five classes : $S_{0} = \{...,-5,0,5,10,...\} \cdots (0)$ $S_{1} = \{...,-4,1,6,11,...\} \cdots (1)$ $S_{2} = \{...,-3,2,7,12,...\} \...
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How many distinct binary strings have Hamming distance at least $k$?

Given length $n$, what's the largest set of distinct binary strings that any two members would have Hamming distance at least $k$? Is there a solution or best strategy to create such a set? For ...
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19 views

Question about Permutations of Separate Groups

So, my question is about what formula you would use if you want to count the combinations and permutations of two separate groups amongst each other. By exact problem is this. I want to find the ...
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0answers
23 views

When the objects are not all distinct, the number of ways to select one or more objects from them is equal to : $(q_{1}+1)(q_{2}+1)\cdots (q_{t}+1) $

Theorem : When the objects are not all distinct, the number of ways to select one or more objects from them is equal to : $$(q_{1}+1)(q_{2}+1)\cdots (q_{t}+1) -1$$ where, there are $q_{1}$ objects of ...
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1answer
21 views

Unsigned Stirling number of the first kind summation

I'm solving a question and I confronted with this summation: $A_n = \sum _{k=1}^n\left(n-k\right)\begin{bmatrix} n\\ k \end{bmatrix}$ Where$ \begin{bmatrix} n\\ k \end{bmatrix}$ is unsigned Stirling ...
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1answer
41 views

Number of different shapes for a full binary tree with height n

I am trying to understand the solution to the following problem: continuation: As per the document, the correct answer for (4) is (a): But I don't get why there's the $-s_{n-1}$ at case 3. $T_1$ ...
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1answer
51 views

3 different books can be distributed among 5 students in an English literature class.

https://imgur.com/a/n2snKwS Question: Find the number of ways in which 3 different books can be distributed among 5 students in an English literature class. Though I can think of different ways to ...
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16 views

How would one enumerate relations that are transitive but not symmetric?

Of late, I have been trying to enumerate various types of relations on a set. Counting transitive relations seems very hard. However, it is known that the number of relations on an n-element set that ...
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1answer
52 views

How to find total number of unique subsets in an array containing duplicates?

We are given an integer array containing duplicates. The array is [2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 5, 5, 5, 5, 7, 7] . After converting the array into a map of <integer, frequencies>, the ...
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37 views

I was giving elitmus test. I encountered this question

How many numbers are possible between 9 to 1000 , such that first digit is greater than second digit? Eg: 12, because 2 is greater than 1. I tried hard but couldn't come up with any solution.
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2answers
31 views

When repetitions in the selection of the objects are allowed, the no. of ways of selecting 'r' objects from 'n' distinct objects is $C(n+r-1,r)$

How to show that : When repetitions in the selection of the objects are allowed, the no. of ways of selecting $r$ objects from $n$ distinct objects is $C(n+r-1,r)$. EDIT It's also known as "Stars ...
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1answer
29 views

If no 3 diagonals of a convex decagon meet at the same point, inside the decagon. Into how many line segments are diagonals divided by their…

Question : If no 3 diagonals of a convex decagon meet at the same point, inside the decagon. Into how many line segments are diagonals divided by their intersections? Answer : The no. of ways of ...
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0answers
12 views

Finding a lower bound on number of edges in cut-set of a directed graph

Let $G=(V,E)$ be a directed graph s.t. $V=\mathbb{Z}_n=\{0,1,\cdots,n-1\}$ where $n$ is odd. Now the directed edges out of $x\in V$ are described. These edges are $x\to 2(x-1)\mod n$, $x\to 2x\mod n$, ...
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2answers
88 views

Show that $(k!)!$ is divisible by $(k!)^{(k-1)!}$

Question : Show that $(k!)!$ is divisible by $(k!)^{(k-1)!}$. Answer : Suppose we've $k!$ objects, where k objects of 1st kind, k objects of 2nd kind,...k objects of $(k-1)!$ kind (why?). And from the ...
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1answer
60 views

How to prove $P(A\cup B\cup C\cup D)$?

I know that $P(A\cup B\cup C\cup D)=$ $$P(A)+P(B)+P(C)+P(D)-P(A\cap B)-P(A\cap C)-P(A\cap D)-P(B\cap C)-P(B\cap D)-P(C\cap D)+P(A\cap B\cap C)+P(A\cap B\cap D)+P(A\cap C\cap D)+P(B\cap C\cap D)-P(A\...
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2answers
34 views

Looking for a simple combinatorics approach to solve a bin-sorting question

I thought stars and bars but it will not work for distinct objects: How many ways to split $4$ identical red balls, $5$ identical green balls and $7$ identical white balls among two bins? I was only ...
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0answers
23 views

Question on combinatorics telling the difference between distribution and division

$\bf{Problem}$ A double-decker bus carry $(u+l)$ passengers, $u$ in the upper deck and $l$ in the lower deck. Find the number of ways in which the $u + l$ passengers can be distributed in the two ...
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0answers
20 views

To prove that, a graph $G$ is bi-colored iff, when in it doesn't have a cycle of odd lengths. [duplicate]

To prove that, a graph $G$ is bi-colored iff, when in it doesn't have a cycle of odd lengths. How to prove this theorem, I need some hints/resource recommendation.
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1answer
57 views

How many shortest paths are there in a brick wall

This is one of the Austrian math competition questions which I am quite sure about. From the top-left corner of this brick wall to the bottom-right corner, how many shortest paths are there? (One can ...
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35 views
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Combinatorics question about cube with vertices marked as + or - 1 and faces marked with the product of 4 vertices [duplicate]

This question is from a practice workbook for a college entrance exam. Let each corner of a cube is represented by +1 or -1 arbitrarily and on each of the six faces of the cube we write the product of ...
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1answer
41 views

Understanding the Bachet's game competition problem

In lectures we were just solving the following problem : So we have n stones and a set S that has given positive natural numbers. Two players take stones from the n stone pile and the one who takes ...
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0answers
31 views

Number of possible 8-character strings

How many 8 character passwords can be made using the 26 letters of the alphabet and the 10 digits (0-9) that don't contain repeated numbers and that contain at least 3 digits?
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2answers
35 views

An elevator starts moving from the ground floor with $8$ passengers, so that all passengers get off the elevator until the sixth floor

An elevator starts moving from the ground floor with $8$ passengers, so that all passengers get off the elevator until the sixth floor. Assuming passengers are the same, then In how many ways is it ...
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1answer
26 views

Circle dividing a set of points [closed]

Suppose there be $2n+3$ points in a plane so that no 4 lie on a circle. Then there exists a circle through 3 points such that $n$ points lie inside the circle and the rest $n$ points lie outside that ...
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1answer
23 views

k distinct books in n identical shelves

How many ways to distribute 6 numbered books into 4 identical shelves? I solved for 3 books into 2 shelves by taking two cases a) Selecting all 3 books and keeping it on one self. ( 3C3- 1 way ) b) ...
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1answer
34 views

Given 2 ≥ colors, a forest with c trees and n amount of vertices: In how different many ways can you paint them? [closed]

So i have this task that gives me a headache. I have 2 ≥ colors and a forest with c trees and n amount of vertices. The question is, if i colorize them, how many options do exists to do that? I would ...
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0answers
62 views

A proof using mathematical induction

By observing the base cases for $t=2,3$, I conjectured the following general inequality which I don't know is true or false: $$\frac{(n_{1}+n_{2}+ \cdot \cdot \cdot n_{t})(n_{1}+n_{2}+ \cdot \cdot \...
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1answer
32 views

A question on permutation and combination with the application of elementry coordinate geometry

In how many shortest ways can we reach from the point (0, 0, 0) to point (3, 7, 11) in space where the movement is possible only along the x-axis, y-axis, and z-axis or parallel to them and change of ...
3
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1answer
31 views

Subgroup of $S_n$ with maximal proportion of derangements

Consider a subgroup $G$ of $S_n$. I'm interested about the proportion of derangements (permutations with no fixing point) in $G$. For example, the cyclic group $C_n$ has $n-1$ derangements and thus ...

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