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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1
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5answers
25 views

Picking (and replacing) among five balls in an urn

An urn contains 5 balls numbered from 1 to 5. A ball is chosen at random and its number is noted the ball is then returned to the urn. this is done a total of 5 times. What is the probability that ...
1
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3answers
52 views

Probability that two numbers differ by one bit

Assuming that t is the bit length of the numbers and that we can pick 2 random numbers (the same number cannot be chosen twice), which is the probability that the two numbers will differ by exactly ...
3
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0answers
11 views

On a metric over m-subsets of [n]

Given an integer $n$, denote the set of integers $\{1,2,\dots,n\}$ as $[n]$. For two $m$-subsets $A$ and $B$ of $[n]$, list their elements in the increasing order as $a_1 < a_2 < \dots < a_m$ ...
-1
votes
3answers
2k views

How many three digits even numbers can we form such that if one of digit is $5$ the following digit must be $ 7$?

How many three digits even numbers can we form such that if one of digit is $5$ the following digit must be $ 7$? I need some ideas on how to proceed on this problem.
17
votes
0answers
243 views

Is this determinant identity known?

Let $A$ be an $n \times n$ matrix that is 'almost upper triangular' in the following sense: entries on and above the main diagonal can be whatever they want, entries on the diagonal just below the ...
1
vote
1answer
37 views

Given $N$, is there a formula for $card( \{(m,n)\, s.t.\, m\cdot n \leq N \} )$?

The formula is also equivalent to : $$ \sum_{m=1}^N \left \lfloor \frac{N}{m} \right \rfloor $$ An interpretation would be to count the discrete rectangles with total area inferior to N. But aside ...
-2
votes
0answers
32 views

Summation of prime multiples less than n [on hold]

How can I sum the following $$ \sum (2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i})) $$ with $$2^x\cdot3^y\cdot5^z\cdot7^w\cdot\prod_1^m(p_i^{a_i}) \le n$$ where $p_i \ge 11$ are list of fixed ...
2
votes
3answers
88 views

High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at ...
1
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2answers
20 views

How many arrangements of the integers 1,2, .., n such that from (ALAN TUCKER Applied Combinatorics)

I was solving a question from alan tucker's applied combinatorial book and got stuck at this question: How many arrangements of the integers 1,2, .., n are there such that each integer differ by ...
0
votes
1answer
53 views

How many ways can this quadrilateral be formed if no two of its vertices are next to each other?

A quadrilateral is formed by joining four vertices of a convex decagon. In how many ways can such a quadrilateral be formed if no two of its vertices are next to each other (that is, no two vertices ...
0
votes
0answers
15 views

Why $|N(P_{i, j})| \cong [0, 1]^n$ as stated in page 21 of HTT?

maybe this is an idiot question, however I could not solve this after thinking for a while. I added the tag about higher categories simply because of the nature of the question, however this is just a ...
2
votes
1answer
133 views

proof of a combinatorial identity

How to prove the following using inclusion exclusion $$ \sum _{k=m} ^{n} (-1)^{k-m} {n \choose k} = {n-1 \choose m-1}$$
2
votes
1answer
52 views

Easiest way to find the 'area of a Venn diagram,' given certain information.

We have a bunch of intersecting regions: $$X_1,\dots, X_n,$$ all with non-negative volume, and we know $V(X_i)$ and $V\left((\cup_{a\in A}X_a)\cap (\cup_{b\in B}X_b)\right)$ for any disjoint ...
1
vote
1answer
50 views

Inviting 4 friends out of 8 for a week such that each friend visits at least once

Dave is inviting 4 friends out of 8 for a week how many possibilities there are such that each friend visit at least once. Let's number the friends for brevity, 1 to 8. This is like asking how ...
1
vote
1answer
209 views

Conflicting answers when using Complements Principle and the Inclusion-Exclusion Principle

The question I'm looking at is: Andy, Bill, Carl and Dave are 4 students on a team of 10. 5 must be chosen for a tournament, how many teams can be picked if Andy or Bill or Carl or Dave must be on ...
2
votes
2answers
224 views

Number of 8 character passwords including numbers and letters without repetition

A password must be created with 8 characters. It can use number or letters, but they cannot be repeated (and letters are not case sensitive so we have only 36 characters). How many passwords are ...
19
votes
17answers
15k views

Applications of the Fibonacci sequence

The Fibonacci sequence is very well known, and is often explained with a story about how many rabbits there are after $n$ generations if they each produce a new pair every generation. Is there any ...
-2
votes
0answers
17 views

mapping of integer to unit circle through function $f(k)=k\theta \pmod{2\pi}$ [on hold]

Let $N$ be a positive integer and $\theta$ an angle in $(0,2\pi)$. Consider the map $f\colon\{0,1,\ldots,N\}\to\text{unit circle}$, defined by $f(k)=k\theta \pmod{2\pi}$. Show that the image of $f$ ...
6
votes
2answers
88 views

Describe and count the set of sequences containing $20$ or $02$

Let $X = \{ 0,1,2 \}$ be a ternary alphabet and denote by $X^*$ the set of finite sequences (i.e. strings) with three symbols. For $w \in X^*$ with $n$ the length of $w$ and $w = w_1 w_2 \cdots w_n$ ...
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votes
1answer
20 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
0
votes
2answers
21 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...
1
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0answers
27 views

How to calculate combinations by drawing out the spaces?

I'm learning about probability on khanacademy. They teach a certain method (they draw out the spaces) to calculate combinations. Two Examples: 1. Take the question "What is the probability to get ...
2
votes
2answers
73 views

a vector inequality and combinatorics related question

This question is a similar restatement of this question which has been recently closed. Let $$A=\{\ (x,y,z)\in\mathbb{N}^3\ |\ 0\leq x,y,z\leq7\}$$ and $$B\subset A \text{ with } ...
2
votes
1answer
26 views

Given $n>0$, let $S$ be a set whose elements are positive integers $\leq 2n$ such that:

S is a set with the property that for all a,b∈S with $a<b$, a doesn't divide b. What is the maximum number of integers that $S$ can contain ? I thought it was the number of prime numbers smaller ...
1
vote
2answers
29 views

Find the total number of matchings in a complete graph with even vertices

I am trying to solve questions from a Walk through combinatorics.., I came across this proof which I was unable prove: Determine the number of perfect matchings for a graph with 2n vertices. I don't ...
1
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0answers
24 views

Count the number of strings containing $ac$ or $ca$ for a fixed length over ternary alphabet $A = \{a,b,c\}$ using rational series

This question is a continuation the one asked here, and which already received good answers. Here I am asking for a solution using rational series of formal languages as suggested by the user J. E. ...
2
votes
1answer
73 views

How many different sums of parts of a vector

The following mathematical puzzle was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
3
votes
2answers
39 views

number of triangles determined by a rectangular grid

Suppose we are given an $m\times n$ rectangular grid of lattice points, such as $S=\{(k,l): 0\le k\le n-1,\; 0\le l\le m-1, \;k,l\in\mathbb{Z}\}$, and we want to determine the number of ...
2
votes
1answer
316 views

Count ways to reach last layer

Consider directed graph which has $N + 2$ layers numbered from left to right by integers from $0$ up to $N + 1$. The leftmost ($0$) and the rightmost ($N + 1$) layers both contain only one vertex ...
2
votes
1answer
47 views

NP combination puzzle (Klotski)

I've written a C++ program to solve sliding puzzles games such as UnblockMe and Car Parking. I'm quite happy about it, since it solves various schemes in less than a second. Recently I fed the game ...
4
votes
2answers
478 views

Counting integer partitions of n into exactly k distinct parts size at most M

How can I find the number of partitions of $n$ into exactly $k$ distinct parts, where each part is at most $M$? The number of partitions $p_k(\leq M,n)$ of $n$ into at most $k$ parts, each of size at ...
2
votes
1answer
60 views

Probability of getting the same vector result

This is part of a mathematical puzzle I was given to me by a friend a while ago and I can't work out how to solve it. Does anyone have any ideas? For a given vector $v \in \{-1,1\}^n$ we consider the ...
3
votes
1answer
37 views

Arrangement counting problem

This is my son's exercise: How many ways that 6 rabbits can be put in 10 cages. I count in 2 different ways: The first rabbit can be in any of 10 cages. Same for the second and so on. So in total, ...
-2
votes
0answers
32 views

combinatorial nullstellensatz [on hold]

I was wondering if there is any trick for selecting the polynomial in Combinatorial Nullstellensatz method by Alon. This could be a powerful tool provided we choose right polynomial.
0
votes
0answers
17 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
0
votes
2answers
33 views

A question on probability of choosing coins

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3 \over 4$ and tails with probability $1 \over 4$. Three coins are chosen from ...
0
votes
1answer
19 views

Isomorphic relation between Catalan representations

There is an unanswered question at MathOverflow: Intersecting Family of Triangulations This article at Wikipedia explains the concept of non-intersecting partitions of a polygon: Catalan number So ...
2
votes
2answers
81 views

Algebraic proof that $\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$

I'm looking for an algebraic proof of this identity for $n, k \in \mathbb{N}$: $$\sum\limits_{i=0}^n \binom{i}{k} = \binom{n + 1}{k + 1}$$ So far, I've turned the left hand side of the equality into ...
3
votes
1answer
25 views

Median of waiting time for $k$-th ace from bridge cards

I can't figure out how to get median of a waiting time from the exercise 36 from W. Feller's book An Introduction to Probability Theory and Its Applications Vol.1 (bold in the quote): ...
2
votes
1answer
33 views

Number of distinct permutations given a character set.

How many distinct three-letter sequences with at least one $T$ can be formed by using three of the six letters of $TARGET$? One such sequence is $T-R-T$. [MathCounts 2005 National Countdown] The ...
14
votes
1answer
240 views
+50

On the inequality $\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$

For all positive integers $n$, $p(n)$ is the number of partitions of $n$ as the sum of positive integers (the partition numbers); e.g. $p(4)=5$ since $4=1+1+1+1=1+1+2=1+3=2+2=4.$ Prove ...
3
votes
1answer
41 views

Choose 8 distinct integers from $\{1, 2,\dots,16,17\}$. Show that the eight contain at least three pairs with a common difference for _any_ choice.

This problem is from the 1999 Canada National Olympiad. I am stuck trying to prove the first part using the pigeonhole principle. Is there a refinement that will allow it to be used more sharply, or ...
0
votes
1answer
29 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
2
votes
1answer
290 views

Number of ways of placing $n$ distinguishable balls in $k$ indistiguishable bins where the maximum size of a bin is $m$

I know that the number of ways of placing $n$ distinguishable balls in $k$ indistinguishable bins is given by the Stirling number of the second kind. But I don't know how to modify it to include the ...
1
vote
1answer
28 views

Argue that $\binom{n}{n_1,n_2,…,n_r} = \binom{n-1}{n_1-1,n_2,…,n_r} + \binom{n-1}{n_1,n_2-1,…,n_r}+…+\binom{n-1}{n_1,n_2,…,n_r-1} $

Argue that $\binom{n}{n_1,n_2,...,n_r} = \binom{n-1}{n_1-1,n_2,...,n_r} + \binom{n-1}{n_1,n_2-1,...,n_r}+...+\binom{n-1}{n_1,n_2,...,n_r-1} $ Each term on the right hand side is the number of ways ...
2
votes
1answer
42 views

How many height arrangements are there for people?

Let's suppose $n$ people of different height stand in line, and the observer (who is smaller than the people in line) looks at them from the side. The observer sees a person unless there is a taller ...
0
votes
1answer
47 views

consider a graph of a gameboard

Consider a graph of a game board. Rounds in the game result in a token moved from a game board location to a game board location, possibly returning to the same one. Let the game board location at the ...
2
votes
2answers
87 views

Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
2
votes
2answers
37 views

How to find the number of all the possible ordered trees with n edges and k leaves?

We know that a tree with n edges have n+1 nodes.So if $|B_{n+1}|$ is the number of all possible ordered trees with n+1 nodes then its true that $C_{n+1} = |B_{n+1}|$ where $C$ is the Catalan ...
1
vote
1answer
68 views

Interesting property of Pascal's Triangle

I was looking at the Pascal's Triangle and saw that for all central numbers in even length row $a \gt 17$, the number $\dbinom{a}{b-2}$ is greater than $\dbinom{a-1}{b}$. This is where $b$ is equal to ...