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. Combinatorics is a ...

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0
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2answers
27 views

Histogram of duplication in n choose k

Imagine having 17 balls to distribute to 4 people. One algorithm for distributing these balls is to give each ball to one out of the four randomly. This means, in an extreme case, it is possible for 1 ...
-2
votes
0answers
40 views

Simplicial homology [on hold]

Let $\Delta$ be the simplicial complex on vertex set [5] whose Stanley-Reisner ideal is $I_{\Delta}=(x_{1}x_{4},x_{1}x_{5},x_{2}x_{5},x_{1}x_{2}x_{3},x_{3}x_{4}x_{5})$. Write the augmented oriented ...
0
votes
1answer
73 views

Generating Functions of Partitions?

Show that $2(1-x)^{-3} [(1-x)^{-3} + (1+x)^{-3}]$ is the generating function for the number of ways to toss $r$ identical dice and obtain an even sum. Workings: I'm not too sure on this problem. ...
0
votes
0answers
64 views

Average Degree of a Random Geometric Graph

A set of $N$ points are distributed randomly on a unit square with uniform distribution. Two points $\mathbf{p}_i$ and $\mathbf{p}_j$ are said to be connected if $\|\mathbf{p}_i - \mathbf{p}_j\| \leq ...
0
votes
1answer
39 views

Probability of $k$ collisions

Say we have $m$ buckets. We select a random bucket and put a ball in it, we repeat this $n$ times. In the end what is the probability of having at least one bucket with exactly $k$ balls? I have ...
0
votes
1answer
62 views

An effcient method of solving a Diophantine equation with 3 variables $Ax+By+Cz=D$?

I'm trying to make an efficient algorithm to find one of the solutions and how many solutions there are to the equation $$Ax+By+Cz=D$$ where $A,B,C,D\in \mathbb Z$ and the range for $x,y,z\in \mathbb ...
6
votes
3answers
587 views

$\tbinom{2p}{p}-2$ is divisible by $p^3$

The problem is as follows: Let $p>3$ be a prime. Show that $\tbinom{2p}{p}-2$ is divisible by $p^3$. The only thing I can think of is that $(2p)!-2(p!)^2$ is divisible by $p^2$ which doesn't help ...
0
votes
0answers
17 views

How many cases can draw diagonals that Applicable 2 above condition?

Imagine A $n$_regular polygon that vertex is named by $1$ to $n$. We know can draw $\frac{(n)(n+3)}{2}$ diagonals in $n$_regular polygon and also know if we want draw Maximum diagonals are not ...
2
votes
2answers
48 views

Determine the number of integer solutions of $x_{1}+x_{2}+x_{3}+x_{4}=32$ where $x_{1},x_{2},x_{3}>0, \space\space 0<x_{4}\leq25$.

Determine the number of integer solutions of $$x_{1}+x_{2}+x_{3}+x_{4}=32,$$ where $x_{1},x_{2},x_{3}>0, \space\space 0<x_{4}\leq25$. My approach is in finding all the solutions with the ...
0
votes
0answers
19 views

Generalized Mobius Inversion formulae

I am having as problem with inverting a relation of the form $f(i)=∑_{j=0}^ig(i,j)h(j)$ I would like to have h in terms of f and g. I know that if my formula was of the form $f(i)=∑_j^ih(j)$ I could ...
1
vote
3answers
41 views

How many ways there are?

I cant solve the following problem. In how many ways we can divide 6 balls between 3 children if every children must receive at least 1 ball. I don't understand the problem. Is it permutations or ...
6
votes
2answers
381 views

Prove that any set of 2015 numbers has a subset who's sum is divisible by 2015

I assume this is correct to any size set, not 2015 in particular... it's obviously true for 2. I know from pen and paper it's true for 3, and 4.... I understand that I should look at the reminders, ...
1
vote
1answer
21 views

Split n balls to k boxes

I have $n$ different balls $(1,2,..., n)$ and $k$ different boxes $(1,2,...,k)$. I want to put all balls to boxes, but if ball i has smaller nuber than j (i < j) than ith ball must be put to box ...
0
votes
1answer
51 views

Generating function $D(x) = (1 + x)(1+x^2)(1+x^3)\cdots$ [on hold]

Let $$D(x) = (1 + x)(1+x^2)(1+x^3)\cdots $$ 1) What is the inverse function of $D(x)$? 2) What sequence is generated by $D(x) $ Please don't vote down, the subject is complicated for me. Sorry ...
2
votes
2answers
99 views

How many even number in a sequence are there?

How many even numbers in the below numbers ? $$\binom{k}{0},\binom{k}{1},\binom{k}{2},\ldots,\binom{k}{k}$$ Worng: Is it true that all of them are odd iff $k$ is odd, and if $k$ is even then ...
3
votes
0answers
35 views

Points Distributed evenly around a circle: how many points are in each region?

A circle of circumference $2$ is split into three arcs of length $\frac{2}{3}$ (so the regions are $[0,\frac{2}{3})$, $[ \frac{2}{3},\frac{4}{3})$, $[\frac{4}{3},2)$, $2$ identifies with $0$) and ...
0
votes
1answer
168 views

Divide 500 into certain group so that all no's 1 to 500 can be found.

500 coins are there. Divide 500 coins into certain bags such that any rupees from 1 to 500 can be found by the combination of the bag's coins. What are the minimum nos of bags ?
7
votes
3answers
642 views

Is every arrangement reachable by shuffling this way?

Suppose we have a vertical stack of $n$ distinguishable coins, each of which is either heads-up or tails-up. Let a shuffle be the following procedure: divide the stack at will into a top- and ...
0
votes
1answer
48 views

Minimum AND operation on subset

Given an array of size N . Let's create all the subsets of this array which contain at least 2 elements. Now, operate AND over the elements of each subset, and store the results in a new array. I ...
1
vote
1answer
95 views

Count pairs with odd XOR

Given an array A1,A2...AN. We have to tell how many pairs (i, j) exist such that 1 ≤ i < j ≤ N and Ai XOR Aj is odd. Example : If N=3 and array is [1 2 3] then here answer is 2 as 1 XOR 2 is 3 ...
0
votes
1answer
222 views

Find sum of all permutations

We call two arrays A and B with length n almost equal if for every i (1 <= i <= n) ...
2
votes
0answers
253 views
0
votes
1answer
110 views

Divide N Hot dogs among M persons

There are N hot dogs and M people and we need to divide the hot dogs equally. Now we need to calculate the minimum number of cuts required to distribute the hot dogs equally. In order to divide the ...
1
vote
1answer
79 views

Count ways to reach Nth row

Given a N*M grid I need to reach last row with following operations : ...
-1
votes
1answer
184 views

Count ways to form isosceles triangles

Their are N persons sitting on a table with N vertices.We need to count the number of isosceles triangles formed such that each vertex of the triangle is a vertex of the table and all persons seating ...
2
votes
1answer
62 views

Multiples of 3 and 5. [on hold]

If we have the Tartaglia(Pascal) triangle in every row which numers are multiples of 3 which are even and which are multiples of 5?
0
votes
1answer
25 views

Partitioning elements into sets

How many ways are there to partition $n$ unique elements into $2$ sets? What about for $k$ sets? I am specifically interested in how to calculate this for varying values of $n$. Moreover, what if ...
1
vote
0answers
18 views

Given a specific rational number, is there a way to find an n and k for the binomial coefficient that will evaluate to it? [duplicate]

Looking at Pascal's triangle, it looks as though all rational numbers can also be expressed as binomial coefficients. Given a rational integer, is it possible to calculate n & k for the binomial ...
1
vote
2answers
32 views

Factorial formula problem [duplicate]

Prove that $(n-r)!(r!)$ divides $ n! $ i know its a factorial formula and it might be easy but i stuck .I tried induction to $n$ or analyzing the factorials but im missing something
2
votes
1answer
291 views

Probability question involving sets of cards

I have an infinite deck built out of sets of 10 cards (in other words 10*n cards). The sets are identical so one '2' is identical to another '2'. A player draws 6 cards. If he draws: any '1' AND a ...
1
vote
1answer
25 views

Number of permutations of $[n]$ with a multiple of $n$ inversions

We have a permutation $\left(a_1,a_2,...,a_n\right)$ of the set $\{1,2,...,n\}$. A pair $(a_i,a_j)$ is said to be an inversion of this permutation if $i<j$ and $a_i>a_j$. Find the number of ...
0
votes
1answer
299 views

I need a formula for how many ways I can choose k balls (two balls each time from the same box) from n boxes?

We have n (can take any value 1,2,3,...) boxes each has the same number of distinct marbles, say b marbles, so the total number of marbles S=n*b. we can choose marbles from boxes with the following ...
5
votes
0answers
92 views

Number of permutations such that adjacent elements don't differ by more than $K$

Given $N$ and $K$, I need to count number of permutations of $1, 2, 3,\ldots, N$ in which no adjacent elements differ by more than $K$. How do I approach this problem?
1
vote
0answers
29 views

Expected size of largest connected component in a random k-out digraph?

Given a digraph with n vertices and kn edges, where each vertex has k out-neighbors randomly chosen at uniform without loops, how would I go about figuring out the expected value of the size of the ...
4
votes
1answer
338 views

What is the number of ways to divide a rectangle into $n$ smaller rectangles line by line?

The original problem was to consider how many ways to make a wiring diagram out of $n$ resistors. When I thought about this I realized that if you can only connect in series and shunt. - Then this is ...
4
votes
2answers
92 views

does $a^2-51b^2=\mp 6$ have a solution for integers?

does $a^2-51b^2=\mp 6$ have a solution for integers? I have tried for many modulos, but could not get much out of them.
3
votes
2answers
876 views

How many perfect squares divide 1!2!3!4!5!6!7!8!9!

What I naturally did was to find the prime factorisation of the product of factorials which is $ 2^{30}3^{13}5^5 7^3 $. Clearly there is 15 unique perfect squares that divide $2^{30}$, 6 unique ...
2
votes
0answers
26 views

Graphs with bounded degree: how many are there?

Can one count the number of undirected (simple) graphs on $n$ nodes with degree at most $d$? Asymptotic bounds would be helpful too.
0
votes
2answers
55 views

The number of nonnegative integer solutions of $x_1+\cdots+x_6=24$ with $x_1+x_2+x_3>x_4+x_5+x_6$

I try to find the number of nonnegative integer solutions of $\begin{align} & {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}+{{x}_{6}}=24 \\ & ...
0
votes
0answers
22 views

How many cases can draw diagonals that Applicable 2 above condition?

Imagine A $n$_regular polygon that vertex is named by $1$ to $n$. We know can draw $\frac{(n)(n+3)}{2}$ diagonals in $n$_regular polygon and also know if we want draw Maximum diagonals are not ...
1
vote
1answer
27 views

Counting with restrictions.

I need help with counting with restrictions, such as in the problem In how many ways can we distribute 13 pieces of identical candy to 5 kids, if the two youngest kids are twins and insist on ...
1
vote
1answer
84 views

Maximise the smallest piece of grid

Given a big rectangular chocolate bar that consists of n × m unit squares. We wants to cut this bar exactly k times. Each cut must meet the following requirements: ...
2
votes
2answers
55 views

A game where starting with 3 boxes, with 10 balls in each, the goal is to remove as many balls as possible following the rules

This is a Norwegian olympiad problem: Peter has three boxes, with ten balls in each. He plays a game where the goal is to end up with as few balls as possible in the boxes. The boxes are each ...
3
votes
3answers
26 views

The probability of selecting both defective items when taking 10 out of 24

The following is a problem from my probability text. A box contains 24 light bulbs, of which two are defective. If a person selects 10 bulbs at random, without replacement, what is the probability ...
0
votes
0answers
20 views

Combinations of inheriting genes with certain variables

Context. The idea is taken from a breeding mechanic of a game similar to inheriting genes. The variables are highlighted in bold and italicized. There are 6 stats from each parent represented by 6 ...
5
votes
4answers
292 views

Probability that 2 appears at an earlier position than any other even number in a permutation of 1-20

Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1,2,3,...,20. What is the probability that 2 appears at an earlier position than any other even number in the ...
0
votes
1answer
25 views

Combinatorial Challenge, alternative solution process.

Problem: "During an election campaign $n$ different kinds of promises are made by the various political parties, $n>0$. No two parties have exactly the same set of promises. While several ...
-6
votes
1answer
327 views

What is the number of positive integers? [closed]

b. In a system of ternary (Base 3) number, with n digits, how many number can be represented? Answer: c. For an n-digit signed 3's complement ternary number (n > 1), what is the number of positive ...
0
votes
0answers
43 views

What is umbral calculus, really? [duplicate]

I've seen this page on umbral calculus as well as wikipedia and and another question asked on this website (What's umbral calculus about?), but I still cannot realize what really umbral calculus ...
1
vote
0answers
27 views

A limit of the hyperfactorial and Barnes G-function

I'm doing some work on the various means (arithmetic, geometric, etc.) of some sequences of binomial coefficients, and I'm having some trouble proving a result regarding a ratio of the Hyperfactorial ...