This tag is for basic 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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1answer
41 views

Is there a set of 69 length-6-sets out of 46 numbers [1..46] so that those length-6-sets “cover” all possible 1035 length-2-sets of 46 numbers?

1.) For this question, we have 46 numbers (balls, cards, whatever): {1,2,3,4 .... 45,46} ======================= 2.) Each length-6-set of 46 numbers ( e.g. {1,2,3,4,5,6} or {1,13,16,17,32,46 } ...
1
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0answers
16 views

minimum number of unit distances required for a unit equilateral triangle

Problem. Suppose we have $n$ points on the plane. Among $\binom{n}{2}$ pairwise distances, there are $e$ number of unit distances. Find minimum $e$ ($e$ as a function of $n$) so that there is a ...
3
votes
1answer
37 views

A small variation of the Magic square problem

Let us consider a $n \times n$ grid squares. We put numbers from $0$ to $n^{2}$ ( note that you can omit any one number from $0$ to $n^{2}$ ) such that sum of elements in each row ,each column and ...
1
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1answer
37 views

paths from from point A to point B with length 8

Question How many paths from point A to point B with length 8 exists that that have even number of negative signs? path example my main problem is that i can't find a good way for counting ...
0
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1answer
38 views

prove $a^2_o-a^2_1+a^2_2-…+(-1)^{n-1}a^2_{n-1}=\frac {1}{2}(a_n+(-1)^{n+1}a^2_n)$

Question if $a_k$ is multinomial coefficient of $x^k$ in polynomial $(1+x+x^2)^n$,where $0\le k\le 2n$,prove: using this equality $(1+x+x^2)(1-x+x^2)=1+x^2+x^4$,show that ...
1
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0answers
37 views

Finding whether a sum of numbers in a set generate another number

I have a set of numbers {a1....an} and another number k. I need to find whether sum of any combination of numbers in the set ...
0
votes
1answer
63 views

Combination Problem with Sofa [on hold]

Suppose we have 5 sofa on room A. in this room, 4 students seated on these sofa. These Strudents go to another room for eating dinner, and after that come back to room A. how many way the students can ...
0
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0answers
53 views

Covering by subsets [on hold]

There is set $A$ with cardinality $2^n$. For every $x \in A$ there is $A_x$ - subset of $A$ with cardinality $2^m$, $x \in A_x$. $M=\{A_x|x \in A \}$. Are there $B \subset A$ with cardinality $\ge ...
1
vote
0answers
29 views

Probability that a subset of a degree-regular graph shares at least a certain number of mutual connections

Consider a set of $n$ vertices of common degree $p$. What is the probability that some subset of $x$ vertices from $n$ share $q$ mutual connections within that group of size $x$? i.e. If we have ...
2
votes
2answers
31 views

Flip cards to get maximum sum

Given N cards where if ith card has number x on its front side then it will have -x on back side and a single operation that can be done only once that is to flip any number of cards in consecutive ...
1
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1answer
44 views

Proof that ordinary multinomial coefficients rise monotonically to a maximum and then decrease monotonically

While most computations of ordinary multinomial coefficients for the following case require recursive summations, I found here a closed-form solution: $$(1+x+x^2+\cdots+x^q)^L = \sum_{a \geq 0} ...
1
vote
1answer
63 views

Will I will be able to sit and watch the movie?

Recently I went to the theater. When I came to buy my $3$ tickets (two friends and I), the machine tells me that there is $18$ seats out of $300$ ($15$ rows of $20$ seats). What is the probability ...
1
vote
1answer
41 views

Find extra work done by Bob

Alice has challenegd Bob game of N puzzle.N puzzle is played on N*N grid with each cell containing distinct numbered tile from 1 to N*N-1 Except one which is empty cell and represented as 0. Move ...
2
votes
1answer
23 views

What are the probability that the first two rows of the class are full?

I was boring in my class. So I ask myself the question: What are the probability that the first two rows of the class are full? Knowing that we're $25$ students in my class and the class have ...
5
votes
0answers
75 views
+100

Parity of sum of Kronecker deltas in a graph

For some fixed $n\in\mathbb N$ let $G$ be a graph on the vertex set $\{1,\dots,n\}$ with a total number of $k$ edges $e_1,\dots, e_k$. For any vertex colouring $c(i)$ of the graph, $\delta_e$ is ...
4
votes
2answers
64 views

Ordered partitions of an integer (with a twist)

I would like to know how to prove (preferably algebraically) that $P_1(2,n)=F_{2n+1}$, where $P_1(2,n)$ is what I define to be the number of ordered partitions of an integer, where the number $1$ has ...
1
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0answers
41 views

The probability that exactly / at-least $k$ numbers are in the correct position [duplicate]

Given a sequence of $[1,\dots,n]$ in random order: Let $P_k$ be the probability that exactly $k$ numbers are in the correct position Let $Q_k$ be the probability that at least $k$ numbers are in the ...
0
votes
1answer
36 views

solving by stars and bars methods

In how many different orders can the people Alice, Benjamin, Charlene, David, Elaine, Frederick, Gale, and Harold be standing on line if each of Alice, Benjamin, Charlene must be on the line before ...
0
votes
1answer
36 views

About permutation with repeated identical elements.

First up, I do know the general solution but somehow am unable to use it to solve this kind of problem. I am simply lost. The problem is like this: ...
2
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0answers
68 views

A combinatorics problem about $n\times n$ square

In a $n\times n$ square, a coordinate arrangement is defined as $n$ cells such that for any row or column, there is only one cell taken. For instance, the arrangement taking the diagonal line is ...
4
votes
1answer
58 views

Simple counting problem

Suppose that you have a box with $n$ balls, from the $n$ balls $k$ are white and $n-k$ are black. Now, sequentially you draw (without replacement) the $n$ balls in groups of $m$ (a natural number that ...
6
votes
1answer
121 views

Can the product of $n$ factorials be $n$ factorial?

Are there any solutions to the equation $a_1!\cdot a_2!\cdots a_n!=n!$ with all variables being integers greater than or equal to $2$?
3
votes
2answers
52 views

Proof $e^n*n!$ is an asymptote of $(n+1)^n$

I would like to prove $\lim_{n\to \infty}e^nn!-(n+1)^n=0$. All I have really done is show $(n+1)^n=\sum_{i=0}^n\frac{n!}{(n+1)^i(i!)(n-i)!}$
3
votes
1answer
38 views

How to calculate the number of integer solution of a linear equation with constraints?

If an equation is given like this , $$x_1+x_2+...x_i+...x_n = S$$ and for each $x_i$ a constraint $$0\le x_i \le L_i$$ How do we calculate the number of Integer solutions to this problem?
1
vote
1answer
27 views

Sum of nth powers and generalized polynomial sum

So this is a 2-part question (both parts I believe are closely related): How exactly does on express the sum $$\sum_{i=0}^{k}{i^n} = Q(n,k)$$ in a closed form For arbitrary positive integers ...
4
votes
1answer
83 views

Permutation Partition Counting

Consider the number $n!$ for some integer $n$ In how many ways can $n!$ be expressed as $$a_1!a_2!\cdots a_n!$$ for a string of smaller integers $a_1 \cdots a_n$ Let us declare this function as ...
0
votes
4answers
102 views

how many words can be formed using all letters in the word EXAMINATION

Assuming any sequence of letters is a word, how many words can we form in such a way that the first two letters are different consonants while the last two letters are vowels?
0
votes
1answer
33 views

What is the minimum number of sign patterns in $\frac n2$ of columns (or rows) of Hadamard matrices?

Given a Hadamard matrix of size $n$, I want to know what is the minimum number of unique sign patterns in any $\frac n2$ columns (or rows). I count a sign pattern and its negation to be the same. My ...
1
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2answers
38 views

Number of $n$ letter words from letters a,b that contain exactly $m$ substrings “ab”

I want to prove the number of $n$ letter words that just have letters a,b that exactly have $m$ "ab" expression is $n+1 \choose ...
0
votes
1answer
27 views

Combination with repition, Representation techniques.

Consider the following Question:A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernick bagels, sesame seed bagels, raisin bagels, and plain bagels. How many ways are ...
2
votes
1answer
25 views

number of ways to put 4 black,4 white,4 red balls in 6 different boxes

The question says:in how many ways we could put 4 black,4 white,4 red balls in 6 different boxes? boxes are distinguishable,black balls are identical,red balls are identical,and white balls are ...
0
votes
0answers
89 views

Question about Combinatorics [duplicate]

I understand that for a problem such as 59C5 there are 5,006,386 possible combinations. Is there a way mathematically to determine exactly how many of the 5,006,386 5-digit combinations will sum to a ...
0
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3answers
85 views

Math Problem on Probability

In the SmallState Lottery, three white balls are drawn (at random) from twenty balls numbered 1 through 20, and a blue SuperBall is drawn (at random) from ten balls numbered 21 through 30. When you ...
-2
votes
3answers
34 views

How can I calculate the total number of possible anagrams for a set of letters?

How can I calculate the total number of possible anagrams for a set of letters? For example: "Math" : 24 possible combinations. ...
2
votes
0answers
28 views

Rotation Algorithim

I have a series of 7 tables and 73 participants in a roundtable discussion. My challenge is to rotate all 73 participants to each of the 7 tables while minimizing the times in which they sit with the ...
1
vote
1answer
41 views

Total number of unique n-element sets from a base of unique elements

I have searched for the answer for this on the site (and on the Internet) and have not found the answer. I do apologize if this is answered and I do not have the vocabulary to ask or search for the ...
2
votes
1answer
36 views

Subjectivity in combinatorics

I found some questions in combinatorics very subjective for example: With the digits $1,2,3,4,5,6$, how many 4-uplas exists (order matters) where the digit 1 is before 4? The solution of this ...
3
votes
1answer
67 views

Upper bound of $S=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$

EDIT: How can I find a good upper bound to this quantity ? $$S_{n,m}=\sum_{k=1}^{P}k!\binom{P}{k}\binom{Q}{k}$$ where $P=\min\{m,n\}$ et $Q=\max\{m,n\}$.
7
votes
1answer
113 views
+50

Graph partition that span a third of edges

Given a graph G is easy to see that we have a partition $V=V_1 \cup V_2$ so that $$e(G[V_1])+e(G[V_2])\leq e(G)/2$$. How can we improve this result showing that we can choose $V_i$ such that ...
4
votes
2answers
69 views

A game with checkers

Alice puts checkers in some cells of a $8 \times 8$ board such that : There is at least one checker in any $1\times 2$ or $2\times 1$ rectangle. There are at least two adjacent checkers in any ...
0
votes
2answers
26 views

How to find a pointset with unique distances

Is there a way to arrange N number of 2D points within a box so that the distances between the points are unique? I have an application where I can measure the distances between points with some ...
0
votes
2answers
24 views

Permutations on the leading diagonal of a matrix

I have an $n\times n$ matrix with only diagonal components which are $\pm 1$. How many of these matrices can I construct? I know this is a basic combinatorics, but I would appreciate some help ...
6
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1answer
69 views

Is it possible to cover a $70\times70$ square with $24$ squares with side length $1,2,3\ldots24$?

Is it possible to cover a $70\times70$ square with $24$ squares with side length $1,2,3\ldots24$?
1
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1answer
38 views

Rational Series VS Algebraic Series

I am reading a paper on combinatorics. It mentions some generating functions are rational series and others are algebraic series. I do not understand the difference, can someone help? EDIT $1$: The ...
0
votes
1answer
42 views

How to calculate the sum of combinatorial numbers

For my work on an almost completely unrelated field I came across the following formula. I know that I have learned that all in high school. But since this is more than 15 years ago in which I never ...
6
votes
5answers
107 views

Finding $\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $

Help me to simplify:$$\binom{n}{0} + \binom{n}{3} + \binom{n}{6} + \ldots $$ I got a hunch that it will depend on whether $n$ is a multiple of $6$ and equals to $\frac{2^n+2}{3}$ when $n$ is a ...
2
votes
3answers
56 views

Distinguishability problem /

How many ways are there to put 6 balls in 3 boxes if the balls are distinguishable but the boxes are not? I'm not quite sure how to approach it, $\frac{3^6}{3!}$ is not an integer. Thanks.
4
votes
7answers
111 views

Classic Counting Problems

Does anyone have some good, classic, counting problems? I want things that are interesting, as well as instructive- more than just compute the number of way to get a flush, etc. (Not that those aren't ...
4
votes
3answers
77 views

Transforming a latin square into a sudoku

Can any $9\times 9$ - Latin Square be transformed into a sudoku by just exchanging rows and columns (it is allowed to mix row- and column-exchanges arbitarily and there is no limit for the number of ...
3
votes
1answer
62 views

Chess Knight problem

Which is the number of all possible combinations of the knights, which are not mutually attack? The black knight may move to any of eight squares (black dots). The white knight in this case is ...