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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2answers
21 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
10 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 $(a_1,a_2\dots a_n)$ ...
1
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1answer
31 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. ...
-3
votes
0answers
17 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.
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votes
0answers
13 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.
2
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0answers
28 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?
3
votes
1answer
108 views

Sum of Catalan numbers

What is $C_1 +C_2 + C_3 +... + C_n$, where each $C_i$ is Catalan number? I want to know if we can bound this sum by some function of $n$. I am looking for an upper bound. For sure it is less than ...
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 ...
2
votes
1answer
49 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 ...
2
votes
0answers
23 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 ...
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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
27 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
52 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 ...
1
vote
1answer
102 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 ...
3
votes
2answers
66 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
55 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
47 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
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2answers
25 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 ...
0
votes
2answers
42 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
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0answers
50 views

Kempe chain color swaps in a partially colored map

Crossposted to: http://mathoverflow.net/questions/179340/kempe-chain-color-swaps-in-a-partially-colored-map Question: In this partially Tait's colored map, using only Kempe chain color swaps (as ...
5
votes
1answer
99 views

$X^A \equiv B \pmod{2K + 1}$

I recently found this problem which asks you to find an algorithm to find all $X$ such that $X^A \equiv B \pmod{2K + 1}$. Is there something special about the modulus being odd that allows us to ...
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?
4
votes
3answers
154 views

Closed form of a sum of binomial coefficients?

I have the following function: $T_n(d)=\sum\limits_{k=\frac{n-d}{2}}^{\lceil \frac{n}{2} \rceil}{k\choose \frac{n-d}{2}}$ ${n \choose 2k}$, where $n,d\in \mathbb{N}^0$, and $n,d$ have the same ...
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: $$ ...
11
votes
3answers
205 views

Deducing correct answers from multiple choice exams

I am looking for an algorithmic way to solve the following problem. Problem Say we are given a multiple choice test with 100 questions, 4 answers per question (exactly one of those four being ...
-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 ...
0
votes
1answer
234 views

Combinatorics and Inversion Sequences

Determine the inversion sequence of the following permutation of $ \{ 1,\ 2, \cdots , 8 \}$. $$ 83476215 $$ I just don't understand how that is converted into a series of numbers 0, 1, and 2 and ...
5
votes
3answers
1k views

How many 7-digit even numbers less than 3000000 can be formed using all the digits 1,2,2,3,5,5,6?

I've somewhat got this question down but I'm only half way. How many $7$-digit even numbers less than $3,000,000$ can be formed using all the digits $1,2,2,3,5,5,6$? So I figured that there's ...
2
votes
1answer
8k views

How can I calculate the number of potential combinations in a password?

If I create a $10$ digit password with the following requirements: At least one uppercase letter A-Z - $26$ At least one lowercase letter a-z - $26$ At least one digits 0-9 - $10$ At least one ...
1
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0answers
45 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 ...
4
votes
2answers
364 views

Linear programming for combinatorics/graph theory

I just went to a graph theory talk talking about various fractional graph parameters (but focusing on one). These were defined using linear programming. A question was asked, "How can we learn more ...
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
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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 ...
1
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1answer
224 views

Probability of a slot having exactly $K$ elements

From this question asked in an interview: Consider a hash table with $M$ slots. Suppose hash value is uniformly distributed between $1$ to $M$. Suppose we put $N$ keys into this $M$-slotted ...
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 ...
1
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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
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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 ...
0
votes
0answers
32 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 ...
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 ...
2
votes
2answers
160 views

If a 3D-cake is cut by $n$ planes yielding the maximum number of pieces, then what is the number of pieces with the cake crust?

It is known that a 3D-cake can be cut by $n$ plane cuts at most into $N$ pieces, defined by Cake Number $N=\frac {1}{6}(n^3+5n+6)$. However, some of the pieces would have a crust of the cake as one of ...
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 ...
4
votes
4answers
230 views

Proof of the identity $2^n = \sum\limits_{k=0}^n 2^{-k} \binom{n+k}{k}$

I just found this identity but without any proof, could you just give me an hint how I could prove it? $$2^n = \sum\limits_{k=0}^n 2^{-k} \cdot \binom{n+k}{k}$$ I know that $$2^n = ...
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 ...
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
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
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) ...