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. Combinatorics is a ...

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Monotonic subsquence in a random permutation

I wish to compute the probability of having a log(n) length monotonic subsequence in a random permutation of {1,...,n}.
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14 views

combinatorics digraph question

A digraph $G = G(V,E)$ on the set of vertices $V$ is a graph where every edge $e ∈ E$ is directed. (Note that double arrows are not allowed in a digraph.) How many digraphs on $n$ vertices are there? ...
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26 views

Proving Combinatorics Statements are equivalent [on hold]

How can I prove that $$\binom{n}{r}\binom{r}{k} = \binom{n}{k}\binom{n-k}{r-k}$$ Based on this how can I then prove that $$ \sum_{k=1}^{m}\binom{n}{k}\binom{n-k}{m-k}=2^{m}\binom{n}{m}$$ Thank you ...
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1answer
28 views

Not-increasing sequence. How to count.

From digits $\in \{1,2,3,4,5,6,7,8,9 \} $ create sequence not-increasing of length $ 5 $. How many are there such sequences ? I have no idea how to deal with it. Please help.
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2answers
21 views

Inclusion and exclusion in combinatorics

You have 15 identical balls and must divide them into 4 drawers stacked on top of each other with the following limitations: You have at least 2 balls in each drawer There will be no more than 5 ...
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2answers
19 views

Rolling dice probability by solving inequlity

I was trying to solve a problem where I have to find the probability of the sum of $\mathcal 3$ rolls of a die being less than or equal to $\mathcal 9$. In order to solve the problem I try first to ...
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0answers
13 views

Probability of a sequence of urn draws having some pair of draws with a minium number of “matches”?

I have $U$ urns. Each urn contains some sequentially numbered balls (not necessarily the same count between urns) $1, 2, 3,... N_u$. I draw one ball from each urn $1, 2, 3,...U$ in turn, and note ...
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26 views

Basic question on appication of Sunflower lemma

A sunflower or $\Delta$-system is a collection of sets $\mathscr{F}$ whose pairwise intersections are all the same set $S$, possibly empty. Elements of the collection of sets $\mathscr{F}$ are called ...
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1answer
62 views

Sum over subsets of $\{1,2,\ldots,n\}$ of terms involving a product over that subset

I'm attempting to perform a sum, using products, using all possible combinations, in a function. How would I go about doing this? (I really need to find something that works.) For example, say I ...
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1answer
30 views

Simplest proof about number of arithmetic sequences in set

Given a set $A = \lbrace1,2,3,\ldots,n\rbrace$, where $n \leq 2^{k}$. What is the simplest way to proof that number of arithmetic sequences with lenght $k$ from set $A$ is $< n^2/2$ ?
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Benefits of combinatorial reasoning?

What I usually do instead of counting something, I form a polynomial whose coefficients count it and go from there. If you had to convince someone why they should learn combinatorial reasoning what ...
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A bound on number of elements less than $n$ of a $B_2[g]$ sequence

Let $S \subset \mathbb{N}$. We say $S$ is of type $B_2(g)$ if the number of representation of the form $n = s_1 + s_2 \ (s_1 \leq s_2)$ is bounded by $g$ for every $n \in \mathbb{N}$. Let $S(n)$ be ...
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1answer
24 views

Ramsey Numbers and edge coloring

Show that for every $k \in\mathbb{N}$ there exists an $n \in\mathbb{N}$, where $n ≤ 3k!$ such that if $K_n$ is coloured in $k$ colours then we can find in $K_n$ a triangle whose edges are of the same ...
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3answers
33 views

Permutations - selection

Give the total number of possible arrangements of 3 letters chosen from the word CALCULUS. The answer is 96, but all I can get is 5P3=60 (permutations of 3 from 5 different elements), or 8P3 adjusted ...
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1answer
47 views

Combinatorics - Without order

You have 10 different types balls to choose from. How many different ways are there to choose 5 balls such that no type of ball appears more than twice. My attempt: Case 1 (selecting different ...
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19 views

Discrete Mathematics; Counting, Summations [duplicate]

Let n ≥ 1 be an integer. Prove that: $$ \sum\limits_{i=1}^n i(\frac{n}{i}) = n \bullet 2^{n-1} $$ I am not sure how to prove this, I think I need to use the derivative of $$(1 + x)^ n$$ any help ...
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21 views

How many permutations of a linear equation

How many strictly positive integer solutions does the equation $x_1+x_2+···+x_n = k$ have? (Hint: Consider the equation $y_1+y_2+· · ·+y_n = k−n$ with variables $y_i \ge 0$.) I believe the ...
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1answer
27 views

Find the total number of functions. [on hold]

Consider the two sets $A=\{1,2,3\}$ and $B=\{1,2,3,4,5\}$. Then find the total number of functions from $A$ to $B$ and also find total number of one to one functions from $A$ to $B$.
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2answers
29 views

A digraph is a graph where every edge is directed. How many digraphs on $n$ vertices are there?

So far I have that between any two vertices (say $j$ and $k$) there are 3 options. there is no edge between $j$ and $k$ there is an edge directed from $j$ to $k$ there is an edge directed from $k$ ...
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42 views

How to count the number of substrings in this combinatorics problem?

Let's say I'm making a string of $A$s and $B$s, where the number of $A$s and $B$s are $a$ and $b$ respectively. A total of $a+b \choose a$ such strings are possible. Now, I wish to know the total ...
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41 views

Eliminating the duplicate counts

Consider a set of $k$ objects and assume that $n$ length strings are to be constructed, where $n \geq k$. I want to count a set of $n$ length strings, with the following restrictions 1. all $k$ ...
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1answer
20 views

compositions of $n$ with $k$ odd parts where all $k$ parts are odd

Here's what i've done so far: $S = N^k$ where $N = \{1,3,5,7,9,\ldots\}$ and $N^k = N \times N \times N\times\cdots$ $k$ times $$\Phi_S(x) = \Phi_{}N_\text{odd}^k(x)$$ $$\Phi_S(x) = (x + x^3 + x^5 + ...
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32 views

Max possible number of sets that have 1 and only 1 member in common

I have a set of 25 things that I want to group into sets of 6, with the following conditions: Every set shares one, and only one, member in common with every other set No object can appear twice in ...
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2answers
34 views

Proving Pascal's identity

So I came across Pascal's identity: Prove that for any fixed $r\geq 1$, and all $n\geq r$, $$ \binom{n+1}{r}=\binom{n}{r}+\binom{n}{r-1}. $$ I know you can use basic algebra or even an inductive ...
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1answer
19 views

A starting lineup consists of 2 forwards, 2 guards and 1 center. How many different starting lineups..

A certain school has $4$ forwards, $4$ guards, $3$ centers and $1$ person who can play as either a forward or a guard. How many different starting lineups can be made? I came up with 2 answers to ...
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2answers
26 views

Binomial coefficient identity $\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$ [duplicate]

I'm having a bit of problems proving the following: $$\sum_{k=1}^n k {n \choose k } = n\cdot 2^{n-1}$$ I always seem to get to the line: $2^{n-1} + 1 = 2^n$ which I know is untrue. Could anyone ...
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1answer
31 views

All the combination of cycles of consecutive numbers

Let say that we have $N$ consecutive number $1,2,...,N$ and we want to find all the possible consecutive number cycles of length $2n+1$. For example: $$\begin{align}&N = 5\\&n = 3\ \ \ \ ...
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1answer
21 views

Randomized Algorithm for finding perfect matchings

I'm stuck on some of the theory in these notes, i'm trying to learn about randomized algorithms in general and am currently stuck on some notes regarding perfect matchings. Here is a link to the ...
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4answers
35 views

Prove using Newton's Binomial Theorem

Let $n≥1$ be an integer. Prove that $$\sum_{k=0}^n k{n \choose k} = n 2^{n-1}$$ Hint: take the derivative of $(1+x)^n$ . I'm assuming that I need to use Newton's Binomial Theorem here somehow. By ...
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1answer
10 views

Equivalence Classes and Relations of Hexagons

Suppose there is a hexagon in the plane. Consider two colorings of the edges of the hexagon equivalent if you can rotate the hexagon so that edges of the same color map to each other. Suppose you ...
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2answers
33 views

Why Can I divide generating function by $x$

In many books on generating functions author performs following operation to shift coefficients of $F(x) = \sum_i f_ix^i$ to the left $${F(x) - f_0} \over x$$ which in can be written as $$(F(x) - ...
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1answer
58 views

Almost every graph is asymmetric?

Here is a question: If i choose at random an isomorphism class of graph(no loops, undirected) on n vertices(with uniform probability on the set of such isomorphism classes), is the probability that ...
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1answer
25 views

Show that given $N$ iid variates $X_i$ uniform on (0,1), $P(\max(\{x_i\} > \frac{1}{2}\sum x_i)$ is $\frac{1}{( N-1)!}$

Given an ensemble of $N$ random uniform variates on $(0,1)$, the probability that the greatest variate exceeds the sum of all the other variates is $\frac{1}{(N-1)!}$. Is there any nice way to prove ...
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37 views

How many expressions can be formed with two commutative and associative functions?

Suppose we have two binary functions $f,g$ which are commutative and associative, i.e., satisfying $$ f(a,b) = f(b,a) \qquad g(a,b) = g(b,a)$$ $$ f(a,f(b,c)) = f(f(a,b),c) \qquad g(a,g(b,c)) = ...
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1answer
25 views

Prove that a sequence can be enumerated using Catalan numbers

This problem is taken from R.P. Stanley’s Enumerative Combinatorics. Give bijective arguments to show that sequences of $n$ $1$'s and $n$ $-1$'s in which the sum of the first $i$ terms is ...
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2answers
32 views

Arranging identical balls in a circle

In how many ways can 4 identical red balls and two identical white balls be arranged in a circle? This is an elementary problem, but many tries have not yet yielded results. I tried by taking the ...
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1answer
23 views

Nearest neighbour algorithm (or so I think).

The algorithm is as follows: Given a graph, we start with some arbitrary vertex, in this vertex the path starts. From a vertex we are at we proceed to a neighbour vertex along some edge, we're keeping ...
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2answers
32 views

placing couples in a circle combinatorics question

In how many ways you can sit n men and n women so that : a) Every man sits near his wife. b) None of the men can sit next to thier wives. I think the answer for A is 2(n-1)!, not sure if it's true ...
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3answers
49 views

How many $6$ digit numbers have their digits in increasing order?

I can calculate the amount of ways you can choose $6$ digits out of $($1,2,3,4,5,6,7,8,9$)$, but this would include combinations where there are $2$ or more of the same digit.
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3answers
41 views

Stirling Numbers Proof

Prove the following: $$\sum\limits_{k=1}^{∞} (−1)^k (k − 1)! S(n,k) = 0$$ Where $S(n,k)$ is the Stirling numbers of the second kind. (Hint: Recurrence Relation) Workings: The recurrence relation ...
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2answers
79 views

Counting the numbers with certain sum of digits.

The question : In how many different numbers between $1$ and $100000000$ have the sum of their digits equal to $45$? I'm thinking about using the stars and bars formula but I'm not sure if it's ...
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2answers
35 views

Find all natural numbers for which $3\binom{2n}{n+1}=2\binom{2n+1}{n-1}$ holds true

I end up getting a quadratic equasion with no natural answers, so I am probably wrong. (Dont know if the tag is right, its part of the combinatorics section in my book)
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3answers
30 views

Combinatorics question about picking a staff

This is the Question : In a building there are 5 men and 5 women. we need to pick representive for the building so that at least one woman and at least one man has to be there. there are no limitions ...
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1answer
27 views

Counting relations question

I have a small question about relation counting, i'm looking for formulas. I know that there is a formula for reflexive and anti reflexive. I'm not sure about the simetric or a-simteric ones, and if ...
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46 views

Why the space of all permutations of a vector (n!) is smaller than the space of all possible permutations of a sorting network?

Imagine you have a vector with 2048 entries. The total permutations are 2048! Now you have a sorting network let us say AKS, the total number of possible results with nlog(n) gates is $2^ {n log (n)}$ ...
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1answer
46 views

A few basic Counting Problems

I don't know if I got these correct. Can someone check for me? How many ways are there to roll a sum of 7 with three standard 6-faced die? There is: 1,1,5 1,2,4 1,3,3 1.4.2 1,5,1 2,1,4 2,2,3 2,3,2 ...
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3answers
275 views

How to check my answer in combinatorics problems

Combinatorics problems (combinations and permutations) are an absolutely maddening subject for me. I can seem to work my way to the answer, provided I already know the correct answer. However, I can ...
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9 views

Explain how lines and points in the 2D plane form an affine plane?

I think I understand the affine transformation, but I just have trouble describing how lines and points in the 2D plane form an affine plane.
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2answers
22 views

How many four digit numbers divisible by five whose digits add up to 6 exist?

I am just learning the basics of combinatorics and my quick answer to this was 22. Though the approach was a bit rough and I sont know how mathematical in nature.
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1answer
30 views

Let n>=2, k>=2. The set of all k-element subsets partitioned into 4 classes: (i) class of subsets containing both 1 & 2, how many k-element subsets?

Sorry for the long title, I'm new here & not sure of the appropriate way to post long questions. The full question is: Let n>=2,k>=2. The set of all k-element subsets of [n] may be partitioned ...