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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0
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2answers
18 views

Probability of a certain outcome

A bag of 14 marbles, 8 red and 6 blue and four marbles are to be chosen at random. a) What is the probability that exactly 2 red marbles and 2 blue marbles are selected? b) What is the probability ...
0
votes
1answer
18 views

Finding out co-linear points

How many triangles with positive area can be drawn on the coordinate plane such that the vertices have integer coordinates $(x,y)$ satisfying $1≤x≤3$ and $1≤y≤3$? It is easy that we have ...
-1
votes
0answers
37 views

proof of a theorem in a paper

I was reading a paper named decomposition of kronecker product of cycles and path into long cycles and paths. In one theorem I have a doubt. I am not getting how the proof of Lemma 1.3 is done. I am ...
2
votes
1answer
20 views

How to simplify?

$\sum_{k=5}^{\infty}\binom{k-1}{k-5}\frac{k^3}{2^{k}} $ The first thing I thought of was the binomial coefficient. So I re-indexed it $$\sum_{j=4}^{\infty}\binom{j}{j-4}\frac{(j+1)^{3}}{2^{(j+1)}} ...
0
votes
0answers
11 views

Find upper bound $\sum_{n=m}^{k} \left[ \binom{b-a}{n-m} \sum ^m_{j=0} …\right]$

I would like to find the upper bound of this expression: $$S_n = \binom{a}{m+1}\sum_{n=m}^{k} \left[ \binom{b-a}{n-m} \sum ^m_{j=0} \frac{\binom{a-m-1}{j} \binom{b-a-n+m}{n-j} } {{b \choose ...
1
vote
0answers
29 views

Count the number of ways n different-sided dice can add up to a given number

I am trying to find a way to count the number of ways n different-sided dice can add up to a given number. For example, 2 dice, 4- and 6-sided, can add up to 8 in 3 different ways: ...
0
votes
0answers
41 views

The probability of random permutation leaving the sequence almost unchanged

So let's say I have $52$ completely different kinds of arenas that get shuffled. What are the chances of getting the exact same sequence if only one arena can be out of order? For example, you could ...
1
vote
3answers
76 views

Simplifying $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}$

In trying to simplify my answer to a problem posted recently, I am trying to show that $\displaystyle\sum_{k=0}^{20}(k+4)\binom{23-k}{3}=8\binom{24}{4}$. I know that ...
0
votes
0answers
12 views

exponential bound on the number of possible clusters at $0$ in $\mathbb{Z}^d$

Let us say that $\mathbb{Z}^d$ is given the usual lattice structure as a graph. I want to know the number of connected induced subgraphs of size $k>0$ that include the vertex $0$. Call this ...
3
votes
0answers
46 views

How to show this expression is always a perfect square?

The number of tilings of an $m \times n$ board with $2 \times 1$ dominoes (each placed either horizontally or vertically on two squares of the board) has been shown to be $$\sqrt{\prod_{j=1}^m ...
2
votes
1answer
38 views

Simplifying a generating function in two variables with two binomial coefficients

I'm trying to to make the below expression simpler, and it would be great if it could be expressed as something like $(x+y)^k$. $$ \sum_{i=0}^k\binom{n+1}i\binom{m+1}{k-i}x^iy^{k-i} $$ The number ...
1
vote
0answers
29 views

Counting apartments in spherical buildings

Is there a formula for the number of apartments in a finite, spherical building? To be specific, is there a formula that depends on the associated Coxeter group and the thickness of the building? ...
0
votes
0answers
11 views

How many Homomorphisms are there from one bounded lattice to another?

for a project that I work on, I need to know how many homomorphisms there are from one finite lattice with 0 and 1 to another. I remember that I already worked it out if one of them is the trivial ...
1
vote
2answers
41 views

Upper bound on $ \binom{a}{m+1}\sum ^m_{j=0} \binom{a-m-1}{j}/\binom{b}{j+m+1}$

Given $a,b,m$ such that $0<2m<a<b$. I would like to find out upper bound of $$S = \binom{a}{m+1}\sum ^m_{j=0} \frac{\binom{a-m-1}{j}}{\binom{b}{j+m+1}}$$ Anyone can help me please? Thank you ...
3
votes
1answer
29 views

Number of possible rectangles from at most N identical squares

I was looking to find the number of distinct rectangles possible from at most $N$ identical squares. (Two rectangles are distinct if one cannot be rotated to obtain another) e.g for $N = 6$ , $8$ ...
0
votes
1answer
20 views

Probability $\sum_{j=n+1}^{2n+1} {M \choose m+1}{M-m-1 \choose j-m-1}/{N \choose j} $

I have a prob. problem: A school has $N$ students in which $M$ students are leader (of each class in school), and $N>M$. There are $2n+1$ balls in the black box including $n+1$ blue balls and $n$ ...
0
votes
2answers
43 views

Combinatorial argument $a(n-a)$ $n \choose a $ = $n(n-1)$ $n-2 \choose a-1$

I can not make sense of this; I am looking for a combinatorial argument that would prove the equivalence of this statement. I can prove it with algebraic manipulation. $a(n-a)$ $n \choose a $ = ...
6
votes
7answers
150 views

How many $10$ digit number exists that sum of their digits is equal to $15$?

How many $10$ digit number exists that sum of their digits is equal to $15$? Additional info: First digit from left is not $0$.we could use any digits from $0$ to $9$. I saw in some ...
-2
votes
1answer
101 views

A Twin Primes Sequence [on hold]

How to prove the relation below and is it enough significant to prove the infinitude of the twin primes? For every twin primes $x,y$ there exists $\alpha,\beta$ positive integers such that ...
-1
votes
0answers
78 views

Find the highest story from which an egg can be dropped without breaking it (lowest average, not worst-case scenario) [on hold]

EDIT: Not looking for worst-case scenario solution, but rather the lowest average. This question has been answered many times for worst-case scenario being 14, I know this already :) PROBLEM: You ...
2
votes
2answers
56 views

Number of ways to put one or more of $5$ books in $5$ bags

In how many ways can we put one or more of 5 books in to 5 bags? Additional info: books are labeled. Bags too. One or more bags can remain empty. Things I have done so far: There are $5$ ...
2
votes
3answers
56 views

Combo Identity: How to prove this using Induction [on hold]

$$ \sum_{n = 0}^{\infty} \binom{n + k}{k}x^n = \dfrac{1}{(1 - x)^{k + 1}} $$ Could someone suggest how I should get started to prove this using induction?
1
vote
3answers
30 views

Choosing $2$ groups of $5$ members and $1$ group of $2$ members from $15$ person

In how many ways can we choose $2$ groups of $5$ members and $1$ group of $2$ members from $15$ person? Additional info: groups are not labeled. Things I have done so far: I know number of ...
1
vote
1answer
22 views

Combinatorics Question about balls in boxes

There are 5 balls numbered 1 to 5, and there are 3 boxes numbered 1 to 3. The question asks in how many distinct ways can the balls be put into the boxes if 2 boxes have 2 balls each and the other box ...
0
votes
1answer
19 views

I'm looking for two euclidean polytopes such that their cartesian product is no longer a euclidean polytope.

I'm looking for two euclidean convex polytopes such that their cartesian product is no longer a euclidean convex polytope. Does such a thing exist? Note here by convex polytope I mean the set $ K ...
5
votes
0answers
42 views

distinguishing family of sets

Call a family $F$ of subsets of $S=\{1,2,\ldots,n\}$ distinguishing if for every two distinct subsets $A,B$ of $S$ there exists $X\in F$ so that $|A \cap X|\ne |B \cap X|$. Show that there exists such ...
1
vote
2answers
33 views

what is the > probability that only one letter will be put into the envelope with > its correct address?

Tanya prepared 4 different letters to 4 different addresses. For each letter, she prepared one envelope with its correct address. If the 4 letters are to be put into the four envelopes at ...
1
vote
1answer
51 views

An upper bound? $\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}{n+1\choose j}{n \choose i}/{2n+1 \choose j+i}$

I would like to find the upper bound (or simplification) of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}{n+1\choose j}{n \choose i}/{2n+1 \choose j+i}$$ where ...
1
vote
1answer
73 views

Choosing 5 of 40 people sitting at a circular table so that between any two are at least three other people

$40$ people sit around a circular table. In how many ways can we choose $5$ people so that between any two of them there are at least $3$ other people? Things I have done so far: This question is ...
0
votes
1answer
55 views

Have any one studied this binomial like coefficients before?

Note that the simillarities of following identities. $\dbinom{n}{r}=\dbinom{n}{n-r}$ $\dfrac{n}{n-r}\dbinom{n-r}{r}=\dfrac{n}{r}\dbinom{n-r-1}{r-1}$ $\dbinom{n}{r-1}+\dbinom{n}{r}=\dbinom{n+1}{r}$ ...
-3
votes
0answers
52 views

Existence of a particular monochromatic sequence from a two colouring of $\mathbb{N}$

The positive integers are colored by two colors. Prove that there exists an infinite sequence of positive integers $k_1 < k_2 < \cdots < k_n < \cdots$ with the property that the terms of ...
0
votes
2answers
59 views

Simplify the expression of binom

Any one knows how to simplify this expression or finding upper bound of this expression: $$\sum_{j=1}^{n+1}\sum_{i=0}^{j-1} a^{j+i} {j+i \choose i}$$ where $0<a<1$ is constant. Thanks a lot.
11
votes
5answers
234 views

Unit circle is divided into $n$ equal pieces, what is the least value of the perimeters of the $n$ parts?

A unit disk is divided into $n$ equal pieces, that is, each piece has area $\dfrac\pi n$. equal "pieces" means equal area Let $l_1, l_2,\dotsc,l_n$ be the perimeters of the $n$ parts, ...
1
vote
2answers
69 views

Number of possible patterns?

Using the following rule: Each column and each row must contain at least one point, how many patterns can a 4x4 grid (thus with 16 possible point positions) generate? (this rule would thus make the ...
4
votes
1answer
49 views

Probability of drawing a run of a specific color from an urn with two colors of balls

I was sent a puzzle involving an urn with 128 white balls and 288 black. If the balls are drawn without replacement until the urn is exhausted, what is the probability that a sequence of 10 or more ...
2
votes
1answer
50 views

What do you call a set whose subsets all have unique sums?

An example would be $\{1, 3, 7\}$, which has subsets with sums $1, 3, 7, 4, 10, 8, 11$. What is this called?
0
votes
0answers
25 views

Simplifying a combinatorial sum

Show that \begin{align} y\sum\limits_{i=1}^dx^iz^i\sum\limits_{j=1}^iq^{i-j}G_d(x,y,q\mid j) = y\sum\limits_{i=1}^d(x^iz^i+\cdots+x^dz^dq^{d-i})G_d(x,y,q\mid i) \end{align} where \begin{align} ...
0
votes
0answers
30 views

Is it true that $k^{\lceil f(k) \rceil}(1+o(1)) = k ^{f(k)}(1+o(1))$? [on hold]

Is it true that $k^{\lceil f(k) \rceil}(1+o(1)) = k ^{f(k)}(1+o(1))$? I think the answer is no?
36
votes
13answers
9k views

Why is it that if I count years from 2011 to 2014 as intervals I get 3 years, but if I count each year separately I get 4 years?

I'm not a very smart man. I'm trying to count how many years I've been working at my new job. I started in May 2011. If I count the years separately, I get that I've worked 4 years - 2011 (year 1), ...
3
votes
1answer
30 views

Number of distinct grids formed

Let $n$ be a positive integer and let $\mathcal{G}_n$ be an $n\times n $ grid with the number $1$ written in each of its squares. In each step we multiply all entries of a row or column is multiplied ...
0
votes
1answer
27 views

To calculate number of combination of sequences having 1 and 2 alternating sequences of R and S.

I have a sequence of 6 letters containing 2 P, 2 R , 1 Q and 1 S. I have PPQ, now I have to add two R and one S in that, these can be placed anywhere. There will be total 60 possible ways to do that ...
1
vote
1answer
30 views

Extracting the coefficient of $x^n$ from a fraction

I need help extracting the coefficient of $x^n$ from a $\frac{1-x}{1-2x}$. So far I have that \begin{align} \frac{1-x}{1-2x} &= \frac{1}{1-2x} - x\frac{1}{1-2x}\\ &= \sum\limits_{k=0}(2x)^k ...
0
votes
1answer
30 views

Upper bound of $\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}$

I would like to find max (or sup.) of the sum: $$S=\sum\limits_{j=1}^{n+1} \sum\limits_{i=0}^{j-1}{n+1 \choose j}{n \choose i}.$$ I found $S\le \frac{1}{\sqrt{\pi n}}.2(n+1).4^n$ but It seems it's ...
1
vote
1answer
12 views

Lexicographical rank of a string with duplicate characters

Given a string,you can find the lexicographic rank of a string using this algorithm: Let the given string be “STRING”. In the input string, ‘S’ is the first character. There are total 6 characters ...
0
votes
0answers
17 views

To determine number of times male and females members of 2 family are arranged alternately in a row [on hold]

To arrange male and female (M,F) of 2 families denoted by P,Q and R,S respectively in such a way that there is exactly one flip (flip means PQ or QP together) in one family and considering the case of ...
0
votes
1answer
39 views

Number of triangles formed by all chords between $n$ points on a circle

We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent. How many triangles exist that their apexes could be on circumference of ...
-1
votes
0answers
53 views

Maximise and operation [duplicate]

Given an array of $n$ non-negative integers $A_1, A_2, \dots, A_N$, find a pair of integers $A_u$, $A_v$, where $1 \leq u < v \leq N$, such that the bitwise-and ($A_u$ and $A_v$) is as large as ...
0
votes
1answer
39 views

Distribution, Combination,Arrangments

How many ways can 25 distinguishable balls be placed in two distinguible boxes? *Order/placing doesn't matter *Only unique combinations accepted (e.g a blue ball weather placed in a box first or last ...
0
votes
1answer
17 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
4
votes
1answer
38 views

What is this sequence of all permutations with gaps permissible

Let there be a sequence $a_1, a_2, a_3,...,a_n$ that represent some actions that you know are required to solve a problem. However, you do not know what order these actions need to be taken to solve ...