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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34 views

Combinations or permutations

I have 3 particles and 5 energy levels (0E,1E,2E,3E,4E). I require all possible ways such that the sum of 3 particles equals 6E. Is there a formula that would enable me to compute the possible ways?
1
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1answer
53 views

How many unique ways can I sum $k$ non-negative numbers to $N$?

I have a similar question but not exactly the same as this. I'm trying to determine the number of unique multisets $S\in \mathbb{N}$ that exist when the members are required to sum to a number $N$. ...
6
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2answers
76 views

Exploring $ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe$, particularly $p = 2$.

I was exploring the fact that $$ \sum_{n=0}^\infty \frac{n^p}{n!} = B_pe,$$ where $B_n$ is the $n$th Bell number. I found this result by exploring the series on wolframalpha and looking up the ...
0
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0answers
19 views

Can you partition a rectangle into exactly 3 congruent non-rectangular parts?

Recently I came upon the following result: Theorem (*): Let $n$ be a positive integer not equal to $1,3,5,7,9$. Then it is possible to partition a rectangle into exactly $n$ congruent non-rectangular ...
2
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0answers
50 views

Why is unit circle not sufficient to bound the partial sums?

I want to find vectors $\textbf{v}_1, \dots,\textbf{v}_n$ in $\mathbb{R}^2$ with that $\sum_{i=1}^n\textbf{v}_i=\textbf{0}$ and $\Vert \textbf{v}_i\Vert\leq 1$ for all $i=1,\dots,n$, such that for ...
4
votes
1answer
63 views

Finite sequence with no two consecutive terms

$\newcommand{\N}{\mathbb{N}}$ Let $n \in \N$, we define $[n] \doteq \{1 , \ldots, n \}$. Consider the following $$ H_n^k \doteq \{ z \in [n]^k  \mid \forall i \in [k-1]: \ z_{i+1} \neq z_i + 1 \} $$ ...
0
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1answer
40 views

Counting and Probability String Length

Consider strings that can be made up from the set $\{a, b, c, d, e, f, \cdots, z, 0, 1, 2, \cdots, 9\}$ 1) How many strings of length 8 contain either the letter '$x$' or '$1$'? 2) What is the ...
0
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1answer
22 views

Unimodality of sequence

I have to show the following: a) was pretty easy to show, however, I am not able to get something useful out of the recursive definition in b) and I have no idea how to approach c). What bijection ...
2
votes
1answer
47 views

How many non prime factors are in the number $N=2^5 \cdot 3^7 \cdot 9^2 \cdot 11^4 \cdot 13^3$.

to find non prime factors in the number $N=2^5 \cdot 3^7 \cdot 9^2 \cdot 11^4 \cdot 13^3$. First I tried finding all the factors by adding 1 to each of the exponents and then multiplying them and ...
6
votes
3answers
274 views

How many permutations

How many permutations $\pi \in S_{2n} $ for which $\exists a\in [2n] $ such that set $\lbrace a,\pi (a),\pi ^2(a),\pi^3(a),... \rbrace $ has exactly $n$ elements. I need help to solve this.
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1answer
57 views

What kind of tree it is? How to solve the problem?

I have a tree with following configuration: n is the number of different vertices v ($0 \lt v \le n$). Each vertice ...
0
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0answers
40 views

about complement of a graph

Let $G$ be a $k-$regular graph on $n$ vertices. we know that if $k\geq n/2$, then $G$ is a connected graph. Now, if we take complement of graph $G$ and denote it as $\bar G$ then $\bar G$ will be ...
2
votes
1answer
24 views

Family of sets without 2 disjunct elements, prove the statement

Suppose, that the $F \subseteq 2^{[n]}$ family of sets doesn't have two disjunct elements. Prove, that there is always an $F' \subseteq 2^{[n]}$ family of sets, which contains $F$, $F'$ has no ...
2
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2answers
33 views

How do you find the the sum of a list of permutations?

If you are given the digits 1, 2, 3 and 4 and then are asked to find the number of different 4-digit numbers you can make (repetition is allowed). We can multiply $4 \times 4 \times 4 \times 4 = 256$ ...
0
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2answers
36 views

Congruence for Stirling Number of first kind $s(n,k)$ when $n$ is prime

Let $s(n,k)$ be the Stirling numbers of first kind: $$\prod_{k=0}^{k=n-1}(x-k) =\sum_{k=0}^{k=n}s(n,k)x^k$$ $p$ is prime $\iff$ for all $k\in\{2,..,p-1\}$, $s(p,k)\equiv0\ mod\ p $ How ...
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1answer
31 views

A question on basic combinatorics. [closed]

I wonder in how many ways $n$ women and $n$ men can be sat down a circular table such that no man sits beside a man and no woman sits beside a woman?
3
votes
2answers
66 views

Counting sequences using Catalan Numbers

Count the number of sequences $a_{1},...,a_{2015}$ such that: $a_{i}\in \{-1,1\}$, and $\sum _{i=1} ^ {2015} a_{i}=7$, and $\sum _{i=1} ^{j} a_i >0$ for every $1\leq j\leq 2015$ I assume we have ...
1
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0answers
57 views

Blocks of consecutive natural numbers

Let a < b be natural numbers. Prove that every block of b consecutive natural numbers contains two distinct elements whose product is divisible by ab. Suppose now a < b < c are natural ...
2
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1answer
36 views

$5$ points on a sphere [duplicate]

Diffuse $5$ points on a sphere. Prove there is a closed half-sphere that has at least $4$ points on it.
1
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1answer
44 views

The greatest number of points of intersection of n circles and m straight lines is-

The question is about combinatorics. I have no idea on how to start solving the problem. Please guide me. $(a) 2mn+ {m \choose 2}$ $(b) \frac{1}{2}m(m-1)+n(2m+n-1)$ $(c) {m \choose 2}+2({n \choose ...
1
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0answers
41 views

Intersecting Family of subsets of size k

Suppose that $k$ divides $n$. Then an intersecting family $F$ of $k$-subsets of an $n$-set $X$ has size at most $n-1\choose k-1$. The prove goes as follows: Let $B$ be the set of all partitions ...
2
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3answers
66 views

The number of choices of 3 kinds of crust and up to 6 distinct toppings

David has a pizza shop. There are 3 kinds of crust and 6 different toppings he can chose from. If customers can have as many toppings as they'd like but may not order double of one topping, how ...
0
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1answer
28 views

For a solution of linear recurrence relation, $\lim_{n\to\infty}a_n^{1/n}$ is a zero of a related polynomial

On page 134 of J.H. van Lint's book A Course in Combinatorics, it says from $a_n=5a_{n-1}-7a_{n-2}+4a_{n-3}$ $(n\ge5)$, we find that $\lim_{n\to\infty}a_n^{1/n}=\theta$, where $\theta$ is the ...
1
vote
1answer
37 views

2x2 grid game problem

A friend of mine is attempting to make a webpage that has a game for a 2x2 grid that is similar to the old North, South, East, West game. I cannot for the life of me figure this out. Essentially, ...
4
votes
4answers
177 views

The even-numbered coefficients of the Maclaurin series of $ \frac{1}{\cos(x)} $ are odd integers.

Let’s consider $ G(z) \stackrel{\text{df}}{=} \dfrac{1}{\cos(z)} $ as the exponential generating function of the sequence of Euler numbers. How can one prove that in the Maclaurin series of $ G $, $$ ...
0
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1answer
22 views

Rewrite the sum of the products by interpretation

By interpreting what the following sum is counting and then counting the same object in a different way, rewrite the following sum as a product of two terms (without any sum): $\sum\limits_{k=m}^n$ ...
0
votes
0answers
20 views

How to find the number of words of length $h$ in a subsets $A$?

Let $L=\{0,1\}^*$ (the set of binary words on $0$ and $1$), Given a tuple of words $(w_1,w_2,\cdots,w_n)\in L^n$ and a function $\sigma:[1,n]\to [1,n]$ define the following set: ...
0
votes
2answers
25 views

Permutations; group of 5 boys, 10 girls. What's the probability the person the 4th position is a boy?

Problem description: A group of 5 boys and 10 girls is lined up in random order -- that is, each of the 15! permutations is assumed to be equally likely. What is the probability that the person in ...
0
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1answer
35 views

Some men and women are randomly assigned seats at a round table and no two persons of the same sex are seated next to each other. Probability of this?

Four women and four men are assigned seats at random at a round table. what is the probability that no two persons of the same sex will be sitting next to each other?
6
votes
0answers
66 views

Riddle: Assigning Students into Groups

Suppose you had a classroom with 25 students. You want to assign 6 homework assignments over the course of the term and for each of these assignments students will work in groups of 5. But you want to ...
1
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0answers
47 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
0
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1answer
44 views

no. of disordered pairs of disjoint subsets

I found this question in a book. The same question has been asked before, but I want a more generalised and rigorous, so to speak, answer. The question reads- " Consider the set $S= \{1,2,3,4\}.$ ...
1
vote
2answers
43 views

Consider all the permutations of the word “ENDEANOEL”

Consider all the permutations of the word "ENDEANOEL" : 1)What is the number of permutations containing the word "ENDEA" ? I can't understand how to approach this problem!! 2)Number of permutations ...
0
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1answer
20 views

Reverse permutation, number of inversions, descents, major index

If $w=a_1a_2...a_n \in S_n $, then let $w^r=a_n....a_2a_1$, the reverse of $w$. Express inv($w^r$), des($w^r$) and maj($w^r$) in terms inv($w$), des($w$), maj($w$), respectively. It from Stanley's ...
2
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1answer
28 views

How do people find the number of ways you can put together a rubiks cube?

Just curious. How do people actually find the number of ways you can put together a rubiks cube? How do you find the number of choices? Do you use the same permutation formula? Insight would be ...
3
votes
3answers
109 views

Is this permutations or combinations?

I am a bit confused. When we use the multiplicative principle are we finding the number of permutations or combinations. An example of using this principle is where I have $5$ shirts $3$ pairs of ...
0
votes
1answer
36 views

Combination formula?

I know there is a formula for finding the different combinations when you are dividing them in groups: $$\binom{n}{r} = \frac{n!}{(n-r)!\, r!}$$ However, what if you just want to find the number of ...
0
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0answers
21 views

Other than the icosahedron in which each vertex has degree 5, is there any triangulation of the sphere that meets the following three conditions?

Every vertex has degree > 3. There is no separating triangle (a triangle with vertices of the graph both inside and outside the triangle). Every vertex-coloring using exactly four colors consists of ...
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2answers
24 views

How many ways are there to arrange three of the letters chosen from the set ABCDE? [closed]

Please show your work. I've been looking at this problem for over an hour now and havn't been able to solve it. Thank you!
0
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1answer
42 views

Number of paths in a graph with infinite nodes

Does a graph with infinite nodes that is not fully connected have a countably infinite or a uncountably infinite number of paths originating from a single node? We are only concerned with paths that ...
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2answers
20 views

Prove that if $k\mid n$ then $p(A_k)={1\over k}$

Let $n$ be a natural number, $n=p_1^{a_1}\cdot...\cdotp_m^{a_m}$. Let us randomly choose a number between 1 and $n$ with a uniform, equal chance. Let us denote the event "The number chosen is ...
0
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1answer
41 views

Generating function of derangements

I am pretty new to the topic of generating functions and I would appreciate if someone could help me out with this problem I have. In the lecture we have proven the following generating function for ...
1
vote
1answer
48 views

What is the probability that you get $i$ on the $i^{th}$ trial?

What is the probability that you get $i$ on the $i^{th}$ trial? Match = Get result $i$ on $i^{th}$ trial. What is the probability of $M = 0,1,2,...,6$ matches when: Note: I'm not asking you to do ...
5
votes
3answers
103 views

Find the coefficient of $x^{30}$.

Find the coefficient of $x^{30}$ in the given polynomial $$ \left(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}\right)^5 $$ I don't know how to solve problems with such high degree.
1
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2answers
43 views

Combinatorics Recurrence relation

Let $h_n$ be a number sequence where $h_n = 3h_{n-1} - 2h_{n-2}$ with $h_0 = 0$ and $h_1 = 1$. Compute the ordinary generating function of $h_n$ and then using the generating function compute a ...
2
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2answers
24 views

Probability of Two Suits within Three cards, within 4 cards

I am trying to calculate what is the probability of the 3 random cards of 52-pack containing at least two of the same suit. I am also trying to do the same for the four card variant (so, the ...
0
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4answers
66 views

Combinatorial Proof of an Instance of the Binomial Theorem

Give a combinatorial proof of the following instance of the binomial theorem. For any positive integer $k$, $(k + 1)^{n}$ = $\sum\limits_{i=0}^{n}$ ${n}\choose{i}$$k^{i}$. I have looked at this for ...
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0answers
28 views

Combinatorics: Password consisting of 13 characters. Must contain at least one odd digit, and at most two even digits. How many passwords?

I'm really trying here. I just need help where to go next. Each character is one of the 10 digits 0, 1, 2, ... , 9 What I have so far is that there are 10^13 possible passwords. I'd have to subtract ...
1
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0answers
32 views

Proving Crapo's Lemma

Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to ...
0
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1answer
20 views

All subsets of nonnegative integers such that $a+2b = n$ has one solution for every positive integer n

My friend tackled this problem awhile ago and gave it to me recently. To reiterate, I am trying to find all subsets $S$ of the nonnegative integers such that the equation $a+2b = n$, where $a$ and $b$ ...