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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1answer
34 views

How many ways can the team be created?

I am doing some old exam questions - and I don't know the answer, can some one calculate the result and show how you did it?
0
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1answer
14 views

How many mappings are there between these two graphs?

Let $P_{20}$ be a path of length 20 like so: $x_0$-$x_1$-$~\cdots~$-$x_{20}$ and $G$ a cycle of order 3. Allegedly there are $3 \cdot 2^{20}$ mappings $P_{20}\rightarrow G$, which I don't quite see. ...
0
votes
0answers
8 views

Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
-2
votes
3answers
45 views

Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$

I am given this: Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$ if: $A \subseteq B$ and $B \subseteq C$ The sets are pairwise ...
1
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0answers
15 views

Rule of Product when choosing multiple items from sets

Suppose you have 4 sets, $S_{1}, S_{2}, S_{3}, S_{4}$ and you want to find out how many ways you can select a combination of A items from set 1, B number of items from set 2, C from set 3 and D from ...
1
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0answers
59 views

Sum involving binomial coefficient and gamma function

I was wondering if anyone has ever seen the following sum: \begin{equation} \sum_{j=0}^{n} \left(-1\right)^{j} \binom{n}{j}\frac{\Gamma\left(\mu+j\right)}{\Gamma\left(\mu+j+n+1\right)} ...
2
votes
2answers
106 views

Splitting a set into two disjoint sets five times, minimizing pairs in the same set

Suppose you have a class of 11 students . I want to split the class into two groups five different ways, minimizing the number of times that any two students are in the same group. In more ...
0
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2answers
23 views

total combinations of divisible sums of $3$

The first $12$ natural numbers are given. Two distinct numbers are selected. What's the probability that their sum is divisible by $3$? This looks very easy. I know answer is $1/3$ but in spite of ...
5
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4answers
63 views

Combinatorial proof of summation of $\sum_{k = 1}^{n-1} {n \choose k}= 2^1 + 2^2 + 2^3 +\ldots+ 2^{n-1}$

I am looking for a combinatorial proof for it. I know how to prove it mathematically. Expanding $(1+x)^n$ and replacing $x$ with $1$ will give me the result but I am not able to explain it ...
10
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2answers
106 views

For any $n^2+1$ closed intervals of $\mathbb R$, prove that $n+1$ of the intervals share a point or $n+1$ of the intervals are disjoint

Stuck on a question from 'Introduction to Combinatorics by Martin J. Erickson'. Q: For any $n^2+1$ closed intervals of $\mathbb R$, prove that $n+1$ of the intervals share a point or $n+1$ of the ...
-2
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2answers
53 views

Which is this series [on hold]

When m = 2, series is 1,2,3,4,5.. ...
0
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1answer
14 views

Finding common ranking of contestants in dance competition

At a dance competition there are a number of contestants and $64$ judges. Each judge ranks the contestants from best to worst, with no ties. For any three contestants $A,B,C$, there do not exist three ...
1
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0answers
26 views

One dimentional random walk

I need to calculate the number of such trajectories of length $n$ (started at $0$ and end at $a$), that for giving $k$ substitute following rule: the maximum isn't greater than $a$, and for every ...
1
vote
0answers
24 views

Show that any vertex $v$ of $P$ is half-integral.

Let $G$ be an undirected graph and define $$P=\{x \in R^{V}: x(u)+x(v) \leq 1 \:\:\text{for all edges}\:\: e=uv,\:\: x \geq 0\}$$ Show that any vertex $v$ of $P$ is half integral.
1
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1answer
21 views

Counting models that satisfy PL sentences

I have an assignment where I need to count the number of models of a certain sort which satisfy a given sentence, and I keep finding that the number of models I count exceeds the total number of ...
0
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0answers
29 views

Is there an arbitrarily large set of naturals so that the sum of each two has exactly $n$ prime divisors? What about an infinite set? [on hold]

Is there an arbitrarily large set of naturals so that the sum of each $2$ has exactly $n$ prime divisors where $n$ is fixed? What about an infinite set? For $n=1$ this is clearly false, what ...
0
votes
1answer
33 views

Two candidates, A & B, are running for president. What is the probability that candidate A beats candidate B?

Candidate A has already garnered 80 votes. Candidate B has already garnered 50 votes. The number of votes a candidate must have in order to win the election is 115. The votes of 5 states are still ...
1
vote
2answers
438 views

Number of binary search trees on $n$ nodes of height up to $h$

How can I find the number of binary search trees up to a given height $h$, not including BSTs with height greater than $h$ for a given set of unique numbers $\{1, 2, 3, \ldots, n\}$? For example, if ...
1
vote
0answers
31 views

Given $5$ points on a sphere, divide the surface into $5$ congruent connected regions containing one point.

There are $5$ points on the surface of a sphere. Is it always possible to divide the surface into $5$ connected congruent regions such that each region contains one of the $5$ points?
1
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0answers
37 views

For any given $k$, show that an integer $n$ can be represented as: $n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$

For any given $k$, show that an integer $n$ can uniquely be represented as: $$n={m_1 \choose 1} + {m_2 \choose 2} + \cdots + {m_k \choose k}$$ where $0 < m_1 < m_2 < \cdots < m_k$. My ...
1
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0answers
21 views

Dividing students into groups with added diversity rule

Could someone help me out for a second, please? See here's the problem: 9 greeks, 17 finns, 7 russians, 11 chinese and 8 swedish students are studying in groups. A group can consist of one or more ...
0
votes
0answers
25 views

Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$.

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
0
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0answers
31 views

2016 AMC 10A #18 — Number of ways to label vertices of acube

Each vertex of a cube is to be labeled with an integer $1$ through $8$, with each integer being used once, in such a way that the sum of the four numbers on the vertices of a face is the same for ...
1
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0answers
16 views

Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$ [duplicate]

Combinatorial proof that $(n-r){n+r-1 \choose r}{n \choose r} = n{n+r-1 \choose 2r}{2r \choose r}$. Typically to combinatorially prove something we need to show that the LHS indeed counts the same ...
12
votes
3answers
459 views

Expected value problem with cars on a highway

There is a very long, straight highway with $N$ cars placed somewhere along it, randomly. The highway is only one lane, so the cars can’t pass each other. Each car is going in the same direction, ...
0
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1answer
91 views
+50

all but one sub-strings within a cyclic string

over $GF(q)$ where $q\in\mathbb{N}$, we build a string of size $q^n-1$. now, how can I show that it is always possible to construct that string so it contains all sub-strings of size $n$ exactly once, ...
0
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0answers
20 views

$r$ balls are randomly distributed into $n$ urns. What's the expected number of urns with $k$ balls?

My text book uses the linearity of the expected value to compute it. It defines a random variable $X_i$ that indicates whether the urn $i$ contains $k$ balls or not. So the asked value is $E[X_1 + X_2 ...
1
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0answers
12 views

Best pattern of cinnamonbuns on a baking tray?

Imagine that i have a 50 x 100 cm baking tray, and i have a load of cinnamonbuns, shaped like a circle with a diameter of 10cm. How do i calculate the best place to place my cinnamonbuns, as the ...
0
votes
1answer
45 views

Combinatorial proof for the identity $\binom{m + n}{r} = \binom{m}{0}\binom{n}{r} + \binom{m}{1}\binom{n}{r - 1} + \cdots + \binom{m}{r}\binom{n}{0}$

Think of a set with $m+n$ elements as composed of two parts, one with $m$ elements and the other with $n$ elements. Give a combinatorial argument to show that $\dbinom{m+n}{r}$ = ...
0
votes
1answer
25 views

Using Pascal's formula to derive another formula

Use Pascal’s formula repeatedly to derive a formula for $\dbinom{n+3}{r}$ in terms of values of $\dbinom{n}{k}$ with $k \leq r.$ (Assume $n$ and $r$ are integers with $n\geq r \geq 3).$ I have a idea ...
0
votes
2answers
280 views

Coin Change Problem - Find the number of ways to make change

I read the solution here http://www.algorithmist.com/index.php/Coin_Change basically to find the number of ways to make change: We are trying to count the number of distinct sets. it says " Since ...
1
vote
1answer
55 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
0
votes
1answer
28 views

Number of permutations with given cyclic structure

If $\sigma$ is a permutation made up by the disjoint cycles $\tau_1, \dots, \tau_r$ (including those of length $1$), we call structure of $\sigma$ $$(l_1, \dots, l_r),$$ where $l_1, \dots, l_r$ are ...
0
votes
2answers
26 views

How many times will the innermost loop be iterated

How many times will the innermost loop be iterated when the algorithm segment is implemented and run? Assume $n$, $m$, $k$, and $j$ are positive integers. ...
2
votes
2answers
7k views

How many solutions are there to the equation $x + y + z + w = 17$?

How many solutions are there to the equation $x + y + z + w = 17$ for non-negative integers $w, x, y, z$ ? I don't know if I'm doing this right, but I guessed that the solution would be ...
0
votes
1answer
47 views

There are $n$ seats in a room. If $n$ people come to the room, what is the probability that $j$ specified people occupy $j$ specified seats?

There are $n$ seats in a room. If $n$ people come to the room, what is the probability that $j$ specified people occupy $j$ specified seats? ($j$ names were tagged on the $j$ seats) $n$ people can ...
4
votes
2answers
40 views

What is the probability that these two objects are of the same color?

We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, ...
1
vote
1answer
489 views

Distinct combinations of non distinct elements

Is there any way to count the number of distinct combinations of a set of objects where some objects may be identical? We have the basic formulas for $nPr$ and $nCr$, and I understand how to modify ...
0
votes
0answers
96 views

Find number of rectangles

There is $N\times M$ grid present with numbering as $1,2,\cdots,NM$ (numbering is done row wise. 1st row will contain number from $1,\cdots,M$, second row will contain $M+1,\cdots,2M$ and so on). ...
1
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6answers
54 views

Probability of getting $5$ heads on $10$ (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely ...
2
votes
1answer
44 views

Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...
2
votes
2answers
43 views

How many bit strings of length 8 begin and end with a 1?

A bit string is a finite sequence of $0$’s and $1$’s. How many bit strings of length $8$ begin and end with a $1$? My answer would be: $2^6$. Because we know, that the bit starts with $1$ and ...
2
votes
1answer
44 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
1
vote
1answer
18 views

Counting “How many ways to choose courses to graduate” with constraints

I have a problem like, "to graduate you must choose 6 out of 20 courses, but at least 2 out of the 6 courses must be a math course. 8 out of the 20 offered courses are math courses. How many choices ...
1
vote
1answer
28 views

show that the maximum degree of the graph is 6

Let p1, p2, . . . , pn be n points in the plane such that the distance between any two points is at least one. Let G = (V, E) be the graph such that V = {p1, p2, . . . , pn} and E = {pipj | distance ...
1
vote
1answer
24 views

5-tuples of n integers

If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which the elements of the 5-tuple are written in decreasing order but are not necessarily distinct? In other ...
0
votes
1answer
35 views

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads I already know that the exact same question has already been answered here But I am trying to solve it ...
2
votes
1answer
34 views

In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, …, 500$ such that one number is the average of the other two?

Here's the question which I'm struggling with - In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, ..., 500$ such that one number is the average of the ...
2
votes
1answer
374 views

Preferences for four ladies and four gentlemen where one proposer receives his/her lowest-ranked choice,…

How can I determine a list of preferences for four ladies and four gentlemen where one proposer receives his or her lowest-ranked choice, and the rest of the proposers receive their penultimate ...
4
votes
2answers
110 views

What am I counting wrong?

EDIT: I made a mistake in the beginning, the second condition has changed. Sorry for this. I'm asked to count the number of sets of 4 elements that satisfy the two following conditions: 1) Each ...