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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How to find the number of permutations with offset restriction

First question. Okay I have this problem that I've been trying to figure out for a while. I'm writing a computer program I need to quickly calculate the permutations of a set with 'n' elements with a ...
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2answers
15 views

Circular permutation, sitting 6 people in a round table

6 people sit down a round table. 4 of them belong to group X and 2 of them belong to group Y. How many ways are there for the 6 people to sit down by taking into account that the 2 people in group Y ...
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5 views

What Boolean matrices are reachable from the NXN identity only by adding columns mod 2?

This problem arose from work on a Boolean software problem. Starting from an $NxN$ identity matrix, the only operation allowed is to add some column i to another column j (mod 2) - i.e. for all $k$, ...
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2answers
103 views

How many 0's are in the end of this expansion?

How many $0's$ are in the end of: $$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$ The answer is supposed to be $1100$ but I have absolutely NO clue how to get there. Any advice?
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0answers
10 views

How many closed paths of length n are circling 0 on the square lattice?

The context is that i am studying a course on percolation and we use a very large bound which is $n * 3^n$. Is there a better one? Thanks
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1answer
28 views

How to find number of subsets

Find the number of all unordered pairs $\{A,B \}$ of subsets of an $8$-element set, such that $A\cap B \neq \emptyset$ and $\left |A \right | \neq \left |B \right |$
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1answer
12 views

Combinatorial Surjection Proof

Let $S(n,k)$ be the number of surjections from an $n$-set to a $k$-set. Prove use a combinatorial proof that: $S(n+1,k) = kS(n,k) + kS(n,k-1)$, where $n \geq k$ Workings: this equation counts the ...
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1answer
22 views

Flipping an unfair coin n times

I’m flipping an unfair coin $n$ times. $\mathbb{P}[X=head]=p$ where $p \neq \frac{1}{2}$. What is the probability “head” appears an even number of times? Thank you in advance for your time an ...
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14 views

a way to compute energy of a flow on transient trees

Let $T$ be a rooted tree that is also a transient electrical network (so effective resistance from root to infinity is finite) but with recurrent rays. (So effective resistance root to infinity along ...
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3answers
55 views

How many 20-digit numbers are there which are formed using only the digits 5 and 7 and divisible by both 5 and 7.

now i realised last digit has to be $5$ and position of $7$ wont affect divisibility. ...
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1answer
13 views

Computing Average Number of Successes When Randomness is Involved

I am attempting to write a program that will compute the average amount of a particular product produced when randomness is involved. Let's say that I am trying to produce some widget. Whenever the ...
2
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1answer
8 views

How to figure out how many possible sequences contain a specific criteria

If a 6-sided die is rolled 5 times and each roll is recorded as an element of set A (|A| will be 5 after all rolls), How many results out of all the possible results will have exactly two 4's as ...
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1answer
28 views

$5$ green dyes, $4$ blue dyes, $3$ red dyes. Number of combination of dyes, which can be chosen taking at least one green and blue dye? [on hold]

$5$ different green dyes, $4$ different blue dyes, $3$ different red dyes. The number of combination of dyes, which can be chosen taking at least one green and blue dye ?
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0answers
23 views

Monotone subsequence in a random permutation

I wish to compute the probability of having a log(n) length consecutive monotone subsequence in a random permutation of {1,...,n} (log with base 2). I'm trying to show it's $\leq1/n$, does it make ...
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1answer
22 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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0answers
7 views

Bound from distinct integer summation

We want to find $r$ positive integers $\{a_i\}_{i=1}^r$ such that of atmost $(s+1)^r$ values obtained from $$\sum_{i=1}s_ia_i$$ where $s_i\in\{0,\dots,s-1,s\}$, we insist on some combination of ...
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2answers
45 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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4answers
44 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
23 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
21 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 ...
3
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1answer
33 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 ...
2
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1answer
71 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
36 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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0answers
33 views

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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0answers
17 views

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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4answers
37 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 ...
5
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1answer
53 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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0answers
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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0answers
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
31 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
33 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 $k$ ...
3
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0answers
46 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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0answers
44 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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0answers
36 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 ...
3
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1answer
21 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 ...
1
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1answer
32 views

All the combination of cycles of consecutive numbers [on hold]

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\ \ \ \ ...
0
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1answer
22 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
34 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) - ...
5
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1answer
60 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 ...
8
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1answer
76 views

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

EDIT: I have posted a generalization of this question to MathOverflow here. Suppose we have two binary functions $f,g$ which are commutative and associative, i.e., satisfying $$ f(a,b) = f(b,a) ...
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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
35 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
24 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 ...