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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2
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0answers
16 views

Combinatorial interpretation of an equality

In a recent project, I came up with the following equality which turned out to be extremely useful for counting conjugacy classes in certain division algebras (I won't go into the details here, it's ...
0
votes
1answer
46 views

Interview Question on Set Theory

Following things are true about the flats (100 in total) in the Royal Society: There are 93 flats which have Wi-Fi connection. There are 62 flats which have landline telephone. There are 85 flats ...
1
vote
3answers
41 views

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same.

If I have 12 books and 12 book spaces, how many ways can I arrange these books? Not all spaces have to be filled. All the books are the same. In other words, putting a book in space 1 and a book in ...
0
votes
0answers
14 views

recursive automata, or recursion mod 2.

Consider the list of length $m$ $(1,0,\dots 0)$ we call this list $l_1$, we now define a sequence of lists recursively, where $l_1$ is the previous list, and if $l_n$ is the list $(a_1,a_2\dots a_n)$ ...
-3
votes
0answers
18 views

There are mn+1 different integers randomly arranged [duplicate]

,then,there either exist m integers arranged in the order that one behind is bigger than one infront,or....I have tried look the order in the view of set theory.
0
votes
0answers
14 views

What does a presentation on block design and Latin squares consist of?

I read the wikipedia pages of both and I just cannot understand these two concepts. I have a presentation on both of these topics next week and I need some headway on both of these topics.
3
votes
1answer
31 views

Tiling squares with L-Trominoes

Is there a simple proof that any square besides a 3x3 square with area divisible by 3 is tileable with L-trominos?
2
votes
1answer
30 views

More computationally optimal way to solve probability of N or more empty buckets given B buckets and A balls

Problem What is the probability of observing N or more empty buckets given B buckets and A balls, if you throw the balls into any of the buckets with equal probability. Python Simulation ...
-1
votes
2answers
24 views

One output for input of $n$-tuples using AND, OR, NOT

Let $B$ be set of $\{0,1\}$ and $B_n$ be the set of all strings of length $n$. How many functions can be constructed from $B_n$ to $B$ using logical operators like AND, OR, NOT. Help $\rightarrow$ ...
0
votes
2answers
28 views

Method to solve probability of chips

A bag contains six chips, numbered 1 through 6. If two chips are chosen at random without replacement and the values on those two chips are multiplied, what is the probability that this product will ...
-1
votes
1answer
56 views

Chess Game with N knights [on hold]

Given an infinite chess board containing N knights. The bottom left corner of the board is labelled as (0,0). Two geeks decide to play the following game on the board : In one turn, a player may move ...
3
votes
2answers
67 views

How many ways can $10$ digits be written down so that no even digit is in its original position

If I have the numbers $0,1,2,3,4,5,6,7,8,9$ written down in that order, how many ways can the $10$ digits be written down so that no even digit is in its original position? It would seem that I can ...
-2
votes
1answer
57 views

Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$

I am unsatisfied with the answers here. (Half of which used algebraic methods despite being advised not to!) Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}$ ...
6
votes
1answer
56 views

The number of combinations $(a,b) \in \mathbb{Z}_n \times \mathbb{Z}_n$ such that $a \cdot b = 0$

This question is about a ring for some chosen $n \in \mathbb{N}$ I wanted to find the number $M_n$ of combinations $(a,b) \in \mathbb{Z}_n \times \mathbb{Z}_n$ can be found such that $a \cdot b = 0$ ...
1
vote
1answer
32 views

Finding Kth string by applying given operations

Given is a matrix of dimension 4*4 in which we are given the result when value along row is computed with value along column. ...
1
vote
1answer
103 views

How to convert a problem to a stars and bars problem?

Continued question from here. With certain questions I have $x_i$ being constrained by various different inequalities, I want to know how to remove these from the problem, to bring me back to a ...
1
vote
3answers
49 views

probability rolling a dice 5 times

I can't solve this problem: What is the probability that, when rolling a dice 5 times, the number of times when you get a 1 or 2 is greater than the number of times when you get a 6. any help?
0
votes
1answer
44 views

How to evaluate $\sum_{k=0}^{n} \alpha^k \binom{n}{k}$?

I am trying to show that the function that satisfies $f^\prime(x)=f(x)$ with $f(0)=1$ behaves in an exponential way (in other words, I want to justify writing it as $e^x$). I need to show that: $$ ...
-5
votes
0answers
17 views

<html5> draw circle by arc and triangle [on hold]

I want if click the retacgle, draw a triangle around the circle. source code like this... but, triangle was not good each of positions.. How can i draw a triangle around the circle like attached ...
1
vote
0answers
47 views

Expected frequency of most frequent die roll

Suppose we have an fair $m$-sided die, and we roll it $n$ times. What is the expected frequency $E(n, m)$ of the most frequently rolled face? If we fix $n$ we can calculate $E(n,m)$ like so. Let ...
2
votes
2answers
37 views

Collision of 8 Digit, Base-36 Numbers

I have an algorithm that generates a random 8 digit, base 36 number with uniform distribution. Thus, this algorithm can generate $36^8$ unique numbers. I run my algorithm 10,000 times, and write ...
2
votes
1answer
52 views

Combinatorics of a game

Suppose there are $n$ people sitting in a circle, with $n$ odd. The game is played in rounds until one player is left. Each round the remaining players point either to the person on their right or ...
0
votes
2answers
16 views

How many different ways of displaying prints

Magda has 6 different prints that she wants to hang on her bedroom wall, but she has room to hang only 2 of them. In how many different ways can she display the prints on her wall? I tried $6 \times ...
1
vote
0answers
40 views

Interesting combinatoral identity

With the help of Mathematica I have discovered a following identity. Let $T>1$ be an integer, $x$ be a real number and let q be a positive even integer and $l=0,1,\cdots,q/2$. The following ...
0
votes
4answers
33 views

Probability: Linear Seating Arrangement

Okay, I'm new at probability and statistics, so please try to answer this as thoroughly as possible and explain why you did everything, from using a specific number to why using factorials and ...
0
votes
2answers
24 views

Given the sizes of various intersections, find the size of the union.

in a certain examination, 72 candidates offered maths, 64 offered English, 62 offered French, 18 offered maths and English, 24 offered maths and French, 20 offered English and French and 8 offered ...
0
votes
3answers
93 views

How to use stars and bars(combinatorics)

How to use the stars and bars method? Say I want to find number of combinations I can get with $x_1+x_2+x_3+x_4=22$ Where $x_i\in\mathbb{N}$ Is this the correct time to apply the method?
1
vote
0answers
21 views

Transforming spanning sub-graphs

I have the following question: Suppose we have a finite graph $G=(V,E)$. Now take two arbitrary spanning sub-graphs, i.e. $G_1 = (V,E_1)$ and $G_2=(V,E_2)$ with $E_1,E_2 \subseteq E$. Suppose we ...
1
vote
0answers
37 views

A combinatorial enumeration problem on graph

Let $G$ be a complete graph of order $n$, we now delete $i$ edges from it, then how many complete subgraphs are there with order $m$ in the rest graph? (You can assume $m\ll n$ and $i\ll m$ if ...
1
vote
1answer
21 views

Graph with small average degree has two vertices of small degree

Suppose $G$ is a graph and its average degree $\epsilon(G) = \frac{2|E(G)|}{|V(G)|}$ is in the interval $0 < \epsilon(G) < 2.$ Then clearly $G$ has one vertex of degree at most $1.$ Reading ...
3
votes
2answers
39 views

Vocal group no two singer stand next to each other?

A vocal group consisting of alf,bill,cal,deb,eve, and fay are deciding how to arrange themselves from left to right on a stage How many way to do this if Alf and Fay are the least skilled singer and ...
1
vote
0answers
23 views

Vocal group couples ordering

A vocal group consisting of alf,bill,cal,deb,eve, and fay (3 boys and 3 girls) are deciding how to arrange themselves from left to right on a stage. How many ways to this if There are 3 couples (Alf ...
1
vote
1answer
12 views

Combination selecting a vocal group

A vocal group consisting of alf,bill,cal,deb,eve, and fay (3 boys and 3 girls) are deciding how to arrange themselves from left to right on a stage. How many way to do this if A. The boys should be ...
3
votes
5answers
111 views

Computing $\sum_{i=0}^{\infty}\frac{i}{2^{i+1}}$

I came across this while trying to solve Google's boys & girls problem, and although I know now it's not the right approach to take, I'm still interested in summing ...
0
votes
0answers
33 views

Proving that number of codes with even weight is the same as number of codes with odd weight for a specific code book

Consider the $[n,n]$ code-book $C_0=\{0,1\}^n$ with $n$ being odd and the codes $c_i \in C_0=[c_1,c_2,...,c_{2^n}]$ being sorted in the ascending order of hamming weight (from $0$ to $n$). Now let's ...
2
votes
0answers
43 views

Density of Pythagorean triples

We define a Pythagorean triple as a triple $<a,b,c>$ such that $a,b,c\in \mathbb N$ and $a^2+b^2=c^2$. In order to avoid duplicates, we say that a triple $<a,b,c>$ is legit iff $b>a$. ...
0
votes
1answer
33 views

Non-punctual Boundary

In the book of Bill Thurston, Three dimensional geometry and topology, there is an exercise to show torus can be partitioned into 7 countries, each on one piece and has common (non-punctual) ...
2
votes
1answer
57 views

Stair flight problem

A stair flight has 10 steps. A kid can move in jumps of 1, 2 or 3 steps. Assume the kid starts on the floor (step 0), and always has to end in step 10 because there is a door that needs to be open. In ...
2
votes
1answer
64 views

What is $\lim_{n\to \infty}\frac{2n \choose {n}}{4^n}$? [duplicate]

What is the result of the following limit? $$\lim_{n\to \infty}\frac{2n \choose {n}}{4^n}$$ since $$\sum_{k=0}^{2n}{2n \choose {k}}=2^{2n}=4^n$$ then $$\frac{4^n}{2n+1}\leq{2n \choose {n}}\leq 4^n$$ ...
1
vote
1answer
29 views

Algorithm for retrieving all the permutations (randomized) for a vector sequence 1…N with only unique values

Here is the problem: I have a vector of $N$ elements long (containing only unique values from $1...N$). I am searching for an algorithm to obtain all the (randomized) combinations possible, where ...
2
votes
3answers
263 views

Probability of dying from smallpox?

A family of four is infected with Variola major. There is a fatality rate of 30%. Calculate the probability that... Here are my attempts, The probability that nobody dies, ...
0
votes
0answers
31 views

Given a particular order how many times will it appear in all the possible permutations it has?

I have $10$ different coloured balls. I'm interested in selling them in packs of $15$ and the order is important. I know there are $10^{15}$ different ways of arranging these balls if I include the ...
0
votes
2answers
52 views

Sequence for number of seating arrangements. [on hold]

I have a problem: Find the number of unique ways to seat $n \in \{2,3,4\}$ guests at a round table. When seating guests at a round table two arrangements are considered the same if each person has ...
1
vote
2answers
26 views

Compensation Question

I want to create a compensation system which takes into account two variables. Lets say I have $1M to distribute among ten employees who produce widgets. I want to compensate each employee by two ...
2
votes
1answer
60 views

How many ways to do choose $\leq 10$ from $5$ sets of $30$ objects.

I have $5$ sets of letters each of size $30$ each. More specifically I have thirty 'a's,'b's,'c's,'d's and 'e's. How many ways can I choose to paint $10$ or less of them? So I seem to have $x_1 + ...
1
vote
2answers
80 views

Number of solutions to $a+b+c+d=14$

Where $a,b,c,d\in\{0,1,2,\ldots\}$. I understand how to find to solution (now), however I'm asking why a particular method I tried failed to work. I imagined laying out $14$ objects in a row, and ...
1
vote
1answer
34 views

For an alphabet of size $N$, how many strings have all of its substrings of length $\geq 2$ unique?

For an alphabet of $N$ characters, how many strings can be formed (including the empty string) so that no substring of length $\geq 2$ appears more than once in the string? The maximum length of such ...
1
vote
0answers
24 views

Counting zeros in a factorial(terminal + zeros in between digits)

The usual counting zeros in a factorial asks to count only the terminal zeros.This question, which also asks to count the zeros that are in between digits,for example, 8! (40320, has a zero between 4 ...
1
vote
2answers
67 views

Pulling aces from a split deck

I have a normal deck of 52. I pull the aces, deal it in to 4 piles of 12, and put an ace in each pile. I shuffle each pile like a monkey on meth. I flip cards from one pile, and when I see its ace ...
0
votes
0answers
13 views

Quotients in exterior products

I just started learning exterior products. The way I understand it, one can associate a subspace with with a bunch of spanning vectors using an alternating multilinear form. The 'k-blade' remains ...