For questions about the study of finite or countable discrete structures, specifically 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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2
votes
2answers
79 views

Find a recursion (combinatorial)

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. ...
0
votes
1answer
20 views

Closed solution to double recursion

I have a problem, where a subproblem is counting how many ways there are to interleave two words from disjoint alphabets while keeping the relative order of the letters within each word. For example, ...
2
votes
2answers
247 views

In how many ways can eight people, denoted $A,B,C,D,E,F,G,H$ be seated about a square table that seats two people on each side?

In how many ways can eight people, denoted $A,B,C,D,E,F,G,H$ be seated about a square table that seats two people on each side? My approach: Since each side of the table seats two people, there are ...
1
vote
1answer
19 views

Amount of match combinations of creating a 5 v 5 team from a pool

This question is inspired by the popular games: Dota 2, Heroes of the Storm and League of Legends; where players have to create two teams of 5 from a pool of "Heroes" in each match. How many ...
0
votes
1answer
28 views

Simplify the formula for the number of distributions leaving none of the $n$ cells empty

I'd like to help with the following problem: $$ \binom{x}{r-1} + \binom{x}{r} = \binom{x+1}{r} \tag{8.6}\label{8.6} $$ 7. Let $A(r, n)$ be the number of distributions leaving none of the n cells ...
0
votes
0answers
16 views

Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? [duplicate]

Question: Which partitions $P$ of $n$ give the row and column sums of some $|P| \times |P|$ $(0,1)$-matrix? Someone comes along and gives us the partition $P=\{2,2,3,3,4\}$ of $14$. How can we ...
2
votes
4answers
97 views

When will Andrea arrive before Bert?

The question was as follows- on any given day, Andrea is equally likely to clock in at work any time from 8:50am to 9:06am. Similarly, Bert is equally likely to to clock in at work at any time ...
0
votes
1answer
29 views

Shortest Path Length as mathematical function/expression

I have a graph (unweighted and undirected) of n vertices. My objective is to express the following constraints as inequalities. The degree of any node should be at least 3. The shortest path length ...
-1
votes
5answers
56 views

Picking (and replacing) among five balls in an urn

An urn contains 5 balls numbered from 1 to 5. A ball is chosen at random and its number is noted the ball is then returned to the urn. this is done a total of 5 times. What is the probability that ...
1
vote
3answers
54 views

Right answer, wrong explanation, probability of grids?

Two unit squares are selected at random without replacement from an $n\times n$ grid of unit squares. Find the least positive integer $n$ such that the probability that the two selected squares are ...
0
votes
0answers
34 views

A residue question in integers

Given $N\in\Bbb N$, is it possible to find $9$ positive integers $A_j,N_i$ with $j\in\{1,2,3\}$, $i\in\{1,2,3,4,5,6\}$ such that following holds? $(1)$ $N\log N < A_j < cN\log N$ at every $j$ ...
0
votes
1answer
28 views

combination of 5 digit numbers

looking at 5-digit number when digits can be $1,2,...,9$ and with repetition, $|\Omega|=9^5$ the event of $5$ distinct digits is $9\times 8\times 7\times 6\times 5$? and the event 2 digits the same ...
0
votes
1answer
32 views

Prove that $\sum_{t \vert n} d^3(t) = (\sum_{t \vert n}d(t))^2$ for all $n \in \mathbb{N}$ [duplicate]

here $d(n)$ counts the number of positive divisors of $n$. I've tried 2 things: Using Bell series. But then again it just showed me that the bell series of the square of a function is not the ...
2
votes
2answers
57 views

Generate all De Bruijn sequences

There are several methods to generate a De Bruijn sequence. Is there a general algorithm to create all unique (rotations are counted as the same) De Bruijn sequences for a binary alphabet of length ...
4
votes
1answer
70 views

Proving a Binomial Identity

Problem $\boldsymbol{25}$ [$\boldsymbol{5}$ Points]: Show that $$ \sum_{k=0}^n\binom{n+k}{k}\frac1{2^k}=2^n $$ Hint: Denote the left hand side by $f(n)$ and prove that $f(n+1)=2f(n)$. Original Image ...
10
votes
1answer
86 views

Asymptotic Behavior of a Sum with Binomial Coefficients

The Problem: Find the asymptotic behavior (with respect to $n$) of the following sum $$\sum\limits_{j = 3}^n \binom{n}{j} \frac{(j - 1)!}{2\cdot n^j}. $$ Where the Problem Comes From: If we ...
1
vote
3answers
116 views

Solving the recurrence relation $T(n) = 2T(n^\frac{1}{2}) + c$

I've been trying to do this for hours. I just don't know how. I'm familiar with recurrence relations in the form of $T(\frac{n}{2})$, but what do you need to do to solve $T(n^\frac{1}{2})$? I've ...
2
votes
0answers
24 views

Mathematics of Magic Squares

I have seen many popular accounts of simple magic squares but I would like to find a proper mathematical background to understanding magic squares. What background knowledge do I need. I am a retired ...
2
votes
0answers
249 views

permutation and combination advanced

I have n sets having values less than 100. I need to find how many arrangements could be made if I pick one element from each set such that in the given arrangement there are no duplicates? NOTE: A ...
3
votes
2answers
90 views

a vector inequality and combinatorics related question

This question is a similar restatement of this question which has been recently closed. Let $$A=\{\ (x,y,z)\in\mathbb{N}^3\ |\ 0\leq x,y,z\leq7\}$$ and $$B\subset A \text{ with } ...
1
vote
3answers
66 views

Probability that two numbers differ by one bit

Assuming that t is the bit length of the numbers and that we can pick 2 random numbers (the same number cannot be chosen twice), which is the probability that the two numbers will differ by exactly ...
3
votes
0answers
20 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
-1
votes
2answers
2k views

How many three digits even numbers can we form such that if one of digit is $5$ the following digit must be $ 7$?

How many three digits even numbers can we form such that if one of digit is $5$ the following digit must be $ 7$? I need some ideas on how to proceed on this problem.
17
votes
0answers
245 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
2
votes
1answer
45 views

Given $N$, is there a formula for $card( \{(m,n)\, s.t.\, m\cdot n \leq N \} )$?

The formula is also equivalent to : $$ \sum_{m=1}^N \left \lfloor \frac{N}{m} \right \rfloor $$ An interpretation would be to count the discrete rectangles with total area inferior to N. But aside ...
-2
votes
0answers
43 views

Summation of prime multiples less than n [on hold]

How can I sum the following $$ \sum (2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i})) $$ with $$2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i}) \le n$$ where $p_i \ge 11$ are list of fixed ...
1
vote
2answers
28 views

How many arrangements of the integers 1,2, .., n such that from (ALAN TUCKER Applied Combinatorics)

I was solving a question from alan tucker's applied combinatorial book and got stuck at this question: How many arrangements of the integers 1,2, .., n are there such that each integer differ by ...
0
votes
1answer
53 views

How many ways can this quadrilateral be formed if no two of its vertices are next to each other?

A quadrilateral is formed by joining four vertices of a convex decagon. In how many ways can such a quadrilateral be formed if no two of its vertices are next to each other (that is, no two vertices ...
2
votes
1answer
133 views

proof of a combinatorial identity

How to prove the following using inclusion exclusion $$ \sum _{k=m} ^{n} (-1)^{k-m} {n \choose k} = {n-1 \choose m-1}$$
2
votes
1answer
53 views

Easiest way to find the 'area of a Venn diagram,' given certain information.

We have a bunch of intersecting regions: $$X_1,\dots, X_n,$$ all with non-negative volume, and we know $V(X_i)$ and $V\left((\cup_{a\in A}X_a)\cap (\cup_{b\in B}X_b)\right)$ for any disjoint ...
1
vote
1answer
51 views

Inviting 4 friends out of 8 for a week such that each friend visits at least once

Dave is inviting 4 friends out of 8 for a week how many possibilities there are such that each friend visit at least once. Let's number the friends for brevity, 1 to 8. This is like asking how ...
1
vote
1answer
209 views

Conflicting answers when using Complements Principle and the Inclusion-Exclusion Principle

The question I'm looking at is: Andy, Bill, Carl and Dave are 4 students on a team of 10. 5 must be chosen for a tournament, how many teams can be picked if Andy or Bill or Carl or Dave must be on ...
2
votes
2answers
233 views

Number of 8 character passwords including numbers and letters without repetition

A password must be created with 8 characters. It can use number or letters, but they cannot be repeated (and letters are not case sensitive so we have only 36 characters). How many passwords are ...
19
votes
17answers
15k views

Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
-2
votes
0answers
20 views

mapping of integer to unit circle through function $f(k)=k\theta \pmod{2\pi}$ [on hold]

Let $N$ be a positive integer and $\theta$ an angle in $(0,2\pi)$. Consider the map $f\colon\{0,1,\ldots,N\}\to\text{unit circle}$, defined by $f(k)=k\theta \pmod{2\pi}$. Show that the image of $f$ ...
6
votes
2answers
90 views

Describe and count the set of sequences containing $20$ or $02$

Let $X = \{ 0,1,2 \}$ be a ternary alphabet and denote by $X^*$ the set of finite sequences (i.e. strings) with three symbols. For $w \in X^*$ with $n$ the length of $w$ and $w = w_1 w_2 \cdots w_n$ ...
-2
votes
1answer
22 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
0
votes
1answer
25 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...
2
votes
1answer
27 views

Given $n>0$, let $S$ be a set whose elements are positive integers $\leq 2n$ such that:

S is a set with the property that for all a,b∈S with $a<b$, a doesn't divide b. What is the maximum number of integers that $S$ can contain ? I thought it was the number of prime numbers smaller ...
1
vote
2answers
33 views

Find the total number of matchings in a complete graph with even vertices

I am trying to solve questions from a Walk through combinatorics.., I came across this proof which I was unable prove: Determine the number of perfect matchings for a graph with 2n vertices. I don't ...
1
vote
0answers
24 views

Count the number of strings containing $ac$ or $ca$ for a fixed length over ternary alphabet $A = \{a,b,c\}$ using rational series

This question is a continuation the one asked here, and which already received good answers. Here I am asking for a solution using rational series of formal languages as suggested by the user J. E. ...
3
votes
2answers
41 views

number of triangles determined by a rectangular grid

Suppose we are given an $m\times n$ rectangular grid of lattice points, such as $S=\{(k,l): 0\le k\le n-1,\; 0\le l\le m-1, \;k,l\in\mathbb{Z}\}$, and we want to determine the number of ...
2
votes
1answer
317 views

Count ways to reach last layer

Consider directed graph which has $N + 2$ layers numbered from left to right by integers from $0$ up to $N + 1$. The leftmost ($0$) and the rightmost ($N + 1$) layers both contain only one vertex ...
2
votes
1answer
50 views

NP combination puzzle (Klotski)

I've written a C++ program to solve sliding puzzles games such as UnblockMe and Car Parking. I'm quite happy about it, since it solves various schemes in less than a second. Recently I fed the game ...
4
votes
2answers
487 views

Counting integer partitions of n into exactly k distinct parts size at most M

How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$? The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at ...
2
votes
1answer
70 views

Probability of getting the same vector result

This is part of a mathematical puzzle I was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
4
votes
1answer
41 views

Arrangement counting problem

This is my son's exercise: How many ways that 6 rabbits can be put in 10 cages. I count in 2 different ways: The first rabbit can be in any of 10 cages. Same for the second and so on. So in total, ...
-2
votes
0answers
33 views

combinatorial nullstellensatz [on hold]

I was wondering if there is any trick for selecting the polynomial in Combinatorial Nullstellensatz method by Alon. This could be a powerful tool provided we choose right polynomial.
0
votes
0answers
18 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
0
votes
2answers
34 views

A question on probability of choosing coins

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3 \over 4$ and tails with probability $1 \over 4$. Three coins are chosen from ...