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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0
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1answer
10 views

permutations vs combinations on slot machines with repeating elements on each reel

For a slot machine with 5 reels where there are repeated elements on each of the reel. Example: Reel 1 [ 1, 1, 2, 1, 3, 5, 6 ] Reel 2 [ 1, 2, 3, 4, 5, 5 ] Reel 3 [ 2, 2, 3, 2, 4 ] Reel 4 [ 1, 2, 3, ...
3
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1answer
49 views

Interesting Combinatorial Identities; e.g. $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$

I came across the following combinatorial identity: $$\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$$ Here's the kind of proof which caught my interest: $\sum_k {n \choose k}^2 = \sum_k {n \choose ...
-7
votes
0answers
35 views

Fundamental principle of counting? [on hold]

How many three-digit even numbers are there such that 9 comes as a succeeding digit in any number only when 7 is the preceding digit and 7 is the preceding digit only when 9 is the succeeding digit? ...
0
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1answer
43 views

Probability a blackjack dealer will bust if you know their score and know the exact deck?

If you know the exact cards left in a deck, and the score of the dealer, how can you calculate the exact probability that they will bust? The dealer behaves as follows: If the dealer's score is less ...
1
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1answer
37 views

How many surjective functions $f: X \to \{1,…,j\}$?

How many surjective functions $f: X \to \{1,...,j\}, |X|=j \cdot k.$ can be defined if they must satisfy: $$ |\{x\in X: f(x)=r\}|=|\{x\in X: f(x)=s\} \forall r,s\in \{1,...,j\} $$ My attempt: From ...
1
vote
1answer
38 views

Number of distinct necklaces using K colors

I have a task to find the number of distinct necklaces using K colors. Two necklaces are considered to be distinct if one of the necklaces cannot be obtained from the second necklace by rotating ...
1
vote
1answer
25 views

maximal matching in graph theory

if we have a graph $G = (V,E)$ and the four values $\beta_1(G)$, $\alpha_1(G)$, $\beta(G)$, $\alpha(G)$, where $\beta_1(G)$: Edge independenth number. The maximal number of independent edges in the ...
5
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2answers
113 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
0
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0answers
15 views

Knight paths on homothetic polyominoes

A while back I made the following conjecture : Let $P$ be an arbitrary polyomino .Let a polyomino be good if there exist a path of a knight on it which passes through each little square exactly once ...
0
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1answer
43 views

How to interpret the Generalized Version of Inclusion-Exclusion Principle

This is a follow-up question on the previous post. Let's say there are $n$ properties which are numbered $1,\cdots,n$. And let $A$ be a set of elements which has some of these properties. Then the ...
1
vote
1answer
24 views

The number of ways to draw boundaries of constituencies, subject to constraints

A state comprises 45 counties arranged as 5 rows, running east and west, of 9 counties each, the nine colums of 5 running north and south. They're connected horizontally and vertically, i.e. ...
1
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1answer
15 views

Numbers written into a square grid

I was working on a problem from The Art and Craft of Problem Solving by Zietz, in the chapter called 'The extreme principle.' Here is the problem: "The integers from 1 to $n^2$ are written into a ...
2
votes
2answers
25 views

Special case on counting in a string of 7 letters

I have the following question: Suppose $S_7$ is the set of all strings of length seven that can be formed with the letters $A, B, C, D, E, F$ and $G$ when repetitions are allowed. How many strings ...
-4
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0answers
46 views

Recall structures made from legos [on hold]

Recall structures made from legos. We do not see these as just one lego brick after another, we see substructure. Try to find some substructure in the following lines of proof. Assume r is in Q. ...
4
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1answer
21 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
1
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1answer
33 views

Is this a binomial or multinomial question?

You can donate to a company: $10$ dollars , $20$ dollars or nothing. In a mall there are $70$% young people and $30$ % old people. $50$% from the old people aren't donating anything. ...
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0answers
21 views

How many ways shuffle $n_1$ and $n_2$ balls when we but them together?

I have $n_1$ white balls and $n_2$ black balls, and I want to know how many ways I can make a distinct arrangement from them. For example , $n_1 = 2$, $n_2 = 1$ then there are three distinct ...
0
votes
1answer
18 views

Counting weakly connected graphs with outdegree of exactly one.

If we count all graphs of $N$ labelled vertices, where each vertex has an outdegree of exactly $1$ with no self-loops allowed, we'll find that there are exactly $(N-1)^N$ of them (for every of $N$ ...
2
votes
2answers
40 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...
0
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1answer
28 views

Numbers of factors of (n)(n+1)/2 is product of exponents?

I was trying to find the number of factors of $n(n+1)/2$, and I read this blog article, and it says that the number of factors of it is the product of its prime factor's exponents with one added to ...
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1answer
57 views

Counting the maximum number of intersections.

Let $n$ be a positive integer. Points $A_1,A_2, \cdots, A_n$ lie on a circle. For $1 \le i <j \le n$, we construct $\overline{A_iA_j}$. Let $S$ denote the set of all such segments. Determine the ...
1
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0answers
35 views

How to Evaluate this Summation to Find a Closed Form

While taking the incomplete Bell Polynomil of $x^a$ i found out that: $$ B_{n,k}^{x^a}(x) = x^{ak-n} \sum_{m=0}^k \frac{(am)!(-1)^{k-m}}{m!(k-m)!(am-n)!} $$ Now, what i am wondering is, what is the ...
3
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0answers
51 views

Stirling transform of $(k-1)!$

While reading about combinatorial mathematics, I found this article about the Stirling transform which caught my attention. So, if I wanted to find the Stirling transform of, for instance, $(k-1)!$, ...
0
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0answers
37 views

Prove the function is nondecreasing

Lets take: $A_1,...,A_n$ family of finite, nonempty sets. Define: $$f(t)=\sum_{k=1}^n\left( \sum_{1\le i_1<...<i_k\le n}(-1)^{k-1}t^{|A_{i_1} \cup ... \cup A_{i_k}|} \right)$$ for $t \in [0,1]$. ...
1
vote
1answer
45 views

Number of elements in discrete $n$-dimensional simplex such that $x_1 \leq \ldots \leq x_n$

For positive integers $n,d$, how many elements are there in the set $S = \{(x_1,\ldots,x_n) \in \mathbb{Z}^n\ |\ 0 \leq x_1 \leq \ldots \leq x_n \wedge \sum_i x_i = d \}$? I'm hoping that the order ...
2
votes
1answer
46 views

How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?

I know that $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}=\frac 12((H_k^s)^2-H_k^{(2s)})$$ and $H_k^s=\sum_{n=1}^kn^{-s}$ But, how would I go about finding identities in terms of the harmonic number like ...
3
votes
5answers
97 views

Why count it this way?

This is a very very elementary problem solving technique I was taught some time back. I have been using it but now looking at it, I find it kinda strange why it should be this way. Typically, the ...
0
votes
3answers
31 views

Combinations and Double Factorials

In a village, there are 10 boys and 10 girls. The village matchmaker arranges all the marriages. In how many ways can she pair off the 20 children, if homosexual marriages (male-male or female-female) ...
0
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2answers
22 views

How many n-permutations have no substrings of the type (j,j+1)?

How many n-permutations have no substrings of the type $(j,j+1)$? $$1\leq j\leq n-1 \text{ and } n\geq 2$$ For example, let n be 5: [3 2 1 5 4] is one of the permutations we have to count. [4 ...
2
votes
1answer
28 views

Combinatorics strategie for order

At the moment I have to deal a bit with Combinatorics but I have some problems with it. Let's say I have following situation: Spend 1500 Euro to 4 people so that everyone has a multiple of 100 ...
2
votes
1answer
39 views

Distribution of K balls in N Cells with limitations

In how many ways can i distribute $k$ balls in $n$ numbered cells with the following limitations: 1.Each cell has different number of balls in it 2.Given each cell has more balls than the cell ...
0
votes
1answer
23 views

Counting functions and stirling numbers

Let S= { f | f: A $\rightarrow$ B, |Image(f)|=k}. |A|=m, |B|=n. where k $ \le n, k \le m $ |S|=$ {n \choose k} $ S(m,k) k!. where S(m,k) are the striling numbers of the second kind. What I can't ...
3
votes
1answer
45 views

An inequality relating to moves to P-positions in Nim

I have been researching this variant of Nim. I have been unable to prove the following claim. What is annoying is that I feel I am missing something really obvious. Does anyone have any ideas on how ...
1
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2answers
32 views

Arrangements in which only two of the three empty chairs are next to each other

While studying, I got stuck on this problem: "Seven identical chairs in a row are to be seated by four students. How many arrangements are there such that the only two of the three empty chairs are ...
1
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1answer
26 views

Generating Finite Groups By Random Premultiplication With Generators

Let $G$ be a finite group with identity $e$ and $S$ be a set which generates $G$. Is it always possible to define a procedure of the form: Start with $x=e$. With probability $p_1$, replace $x$ with ...
0
votes
3answers
52 views

Proof for number of completely odd and even subsets.

While studying, I read this: "A subset of integers $1,2,...,n$ has the property that the sum of its members is odd. The number of such subsets is $2^{n-1}$." I also read this: "A subset of integers ...
1
vote
1answer
44 views

What is the sum of all $k$ values?

In an urn there are a certain number (at least two) of black marbles and a certain number of white marbles. Steven blindfolds himself and chooses two marbles from the urn at random. Suppose the ...
1
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0answers
15 views

Available lists of all latin squares up to order 5?

There are available online lists of the number of all latin squares up to order 11, e.g.: https://oeis.org/A002860. For a permutation-based test of a latin square design, one option is to fit the ...
1
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4answers
47 views

Proof for number of ways to select k non-consecutive elements from n consecutive terms. [duplicate]

While studying, I found a formula that found the number of ways to select k non-consecutive elements from n consecutive terms, not necessarily the first n consecutive terms, but any n consecutive ...
4
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0answers
28 views

Proving that the intersection of two closed sets is closed in a matroid

I am stuck on a little homework problem I have. Here, $M$ is a matroid with rank function $R$. I am given this definition: In a matroid $M$, a set $A$ is closed if $R(A \cup e) > R(A)$ for all $e ...
2
votes
1answer
25 views

Tuples in cartesian product without duplicates

I have $n$ sets $S_1,\ldots,S_n$ and I would like to count the number of tuples $(i_1,\ldots,i_n)\in S_1\times\cdots\times S_n$ such as $i_h\neq i_k$ $\forall h,k\in \{1,\ldots, n\}$. Is there a ...
2
votes
0answers
44 views

Unique ways to distribute k1, k2, .. colored balls into n boxes uniquely

Example: Uniquely distribute 2 Red Balls and 4 Blue Balls into 3 boxes: [B][BB][RRB] [B][BBB][RR] [B][R][RBBB] [B][RB][RBB] [BB][R][RBB] [BBB][R][RB] Answer: ...
6
votes
2answers
60 views

$\sum_{k=1}^n \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$

(Own) Let $n,m$ be positive integers such that $m>n$. Prove that $$\sum_{k=1}^n \sum_{a_1+a_2 + \cdots +a_k=n} \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$$ ...
1
vote
0answers
30 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
1
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1answer
50 views

Name of the numbers defined by $T(p,q) = T(p-1,q) + T(p,q-1)$?

I came across these numbers : $$ T(p,q)= \sum_{k=0}^{q-1} {p+k-1 \choose p-1} + \sum_{l=0}^{p-1} {q+l-1 \choose q-1} \quad p,q \in \mathbb{N} $$ While trying to solve this recurrence relation : $$ ...
4
votes
2answers
69 views

Simplifying a combinatorial expression

Find \begin{eqnarray} \sum_{i=1}^{k-1}i(2k-2-i)\binom{2k}{2i+1} \end{eqnarray} I know how to find $\sum_{i=1}^{k-1}a_i\binom{2k}{2i+1}$ if $a_i$ is linear in $i$, but got stuck when $a_i$ is ...
4
votes
1answer
97 views

A Combinatorial Sum!

Is there a closed form formula for the following sum \begin{equation} F(x;n,m)=\sum_{k=0}^{\min\{n,m\}} {n \choose k}{m \choose k}k!\ x^{k}=n! \, m!\sum_{k=0}^{\min\{n,m\}}\frac{1}{k!(n-k)!(m-k)!} ...
-1
votes
1answer
43 views

Calculating distance between two squares of a board

Given an $n\times n$ board, for example a chess board 8x8, with the squares ordered in a Little-Endian Rank-File Mapping. Is there a direct way to calculate the distance between two squares using ...
0
votes
0answers
34 views

Converting base 10 to base 52 using a bijective function [migrated]

I was recently asked in an interview the following question: "How would you design a URL shortener?" My response was to store the URL into a database which provides a unique key of maximum length 10 ...
2
votes
3answers
36 views

Probability theory combinatoric problem

A total of $n$ bar magnets are placed end to end in a line with random independent orientations. Adjacent ends with equal polarities repel each other, and adjacent ends with opposite polarities ...