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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0answers
9 views

Remove minimal number of elements

Given the numbers $ 1,2,..,2n + 1 $ , $ n > 0$ , remove as few numbers as possible so that among the remaining numbers no number is equal to the sum of two other numbers. After removal of first ...
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1answer
41 views

How many 10 digit numbers are there so the sum of the digits is $2$?

How many 10 digit numbers are there so the sum of the digits is $2$? $abcdefghij$ is the 10 digit number. By default, $a=1$ is a must. $= 1bcdefghij$ Now we need: $bcdefghij = 1$ How can I solve ...
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2answers
17 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 ...
0
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1answer
26 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 ...
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0answers
17 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
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1answer
19 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 ...
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3answers
35 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 ...
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0answers
11 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 ...
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0answers
14 views

Question regarding a proof of the Combinatorial Nullstellensatz

N. Vishnoi has provided a slick proof of the combinatorial nullsetellensatz at http://research.microsoft.com/en-us/um/people/nvishno/site/publications_files/valon.pdf . The part that I am not ...
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2answers
43 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 ...
6
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3answers
618 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 ...
3
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0answers
30 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
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1answer
50 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 ...
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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
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0answers
17 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
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2answers
31 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
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1answer
60 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?
1
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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 ...
4
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0answers
72 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?
2
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0answers
25 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.
6
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2answers
370 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
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1answer
25 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
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0answers
22 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 ...
3
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3answers
25 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 ...
2
votes
2answers
54 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 ...
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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 \\ & ...
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votes
0answers
31 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
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 ...
2
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0answers
27 views

The annihilator number of $S/I$ [on hold]

Let $S=K[x_{1},x_{2},...,x_{n}]$and $I$ be the strongly stable ideal of $S$. Compute the annihilator number of $S/I$ with respect to the almost regular sequence $x_{n},x_{n-1},...,x_{1}$. Note that ...
0
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1answer
22 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 ...
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0answers
18 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 ...
1
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0answers
25 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 ...
2
votes
1answer
21 views

Given a graph on $n$ vertices find the maximum amount of edges so it can be colored with no monochromatic $K_m$

I invented a problem and I wanted to share :What is the maximum amount of edges a graph on $n$ vertices can have if it can be edge-colored with $k$ colors so that it does not have a monochromatic ...
0
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1answer
38 views

In how many ways can the word “WORD” be rearranged so that no letter is in its original position?

In how many ways can the word "WORD" be rearranged so that no letter is in its original position? The answer is $9$, but what is the formula for it?
2
votes
3answers
52 views

Proof by induction, binomial coefficient

I have to make the following proof: $${\sum\limits_{k=1}^n}{k}{n\choose k} = n2^{n-1}$$ Base case, $n = 1$: $${\sum\limits_{k=1}^{1}}{k}{1\choose k} = 1 = 1\cdot2^0=1$$ Inductive Hypothesis: for ...
1
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2answers
51 views

Lottery based counting problem based on uniqueness and monotonicity

I was solving this problem and have prepared a solution here. Problem summary: Consider choosing Blank number of integers from 1 to ...
0
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1answer
25 views

How restrictions reduce the number of possible arrangements

A company has five departments. The company is establishing a board consisting of five members that represent a distinct department each. Suppose that every employee is a candidate to represent his ...
4
votes
4answers
63 views

Finding all possible combination **patterns** - as opposed to all possible combinations

I start off with trying to find the number of possible combinations for a 5x5 grid (25 spaces), where each space could be a color from 1-4 (so 1, 2, 3, or 4) I do ...
4
votes
2answers
388 views

Where can the knight be?

The answer is 33. I get $24$. Because of $8 \cdot 3 = 24$? How can I do this using combinatorics?
1
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2answers
43 views

Combinatorics using a geometric diagram

How can I do this without trial-and-error? It has something to do with a triangle and summing the next row?
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1answer
33 views

How many possible paths?

The answer is $32$. Its supposed to be $2^5$ but I do not see how you get that? The way I see it, there are $5$ ways to go up and $5$ ways to go right, total ways = $5x5= 25$
1
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1answer
26 views

Maximal Multiplication of All Possible Summands

I have recently got interested in the following problem: Give a decomposition of a natural number to natural summands whose multiplication is maximal. I have tried to solve this problem, and ...
0
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1answer
29 views

An interesting mathematics task.

Find the number of different ways of arrangement of all natural numbers from 1 to 9 inclusive, one in table cells measuring 3 by 3 such that the sum of the numbers in each row and each column are ...
2
votes
2answers
52 views

Counting bit strings of length 10 contain either 5 consecutive 0's or 5 consecutive 1's

How many bit strings of length 10 contain either five consecutive 0's or five consecutive 1's ? My Solution: for 5 consecutive 0's After we have filled 0's from $1^{st}$ position we have 2 ...
0
votes
1answer
28 views

Probability of getting an average of 3 or more by rolling 4 sided die twice

What I understood is the sample mean of two rolls of all sample space(16) as given below: ...
0
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0answers
12 views

Ways of partitioning n points into some cubes

Assume there're $n$ fixed points in $\mathbb{R}^d$ contained in a ball with radius $M$,and you can partition the space by cubic grid with cube's edge length $h>\epsilon$. How many different ways of ...
1
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2answers
36 views

Pairs of integeres for which the arithmetic mean exceeds the geometric mean exactly by $2$

Suppose $0<x<y<2015$ are integers. How many pairs of $x$ and $y$ are there for which the arithmetic mean exceeds the geometric mean exactly by $2$? Progress Obtained the equation ...
6
votes
1answer
70 views
+50

A set of integers whose elements all divide $2015^{200}$ but do not divide each other

Let $S$ be a set of natural numbers,such that each element divides $2015^{200}$ but for no two elements $a$ and $b$, $a|b$. Find the maximum number of elements in $S$ . $2015^{200}=(5\cdot ...
1
vote
2answers
28 views

Number of 5 letter words with at least one double letter

How many 5 letter words have at least one double letter, i.e. two consecutive letters that are the same? Answer is: $26^5 – 26*25^4 = 1,725,126 $ But how can i solve? I don't understand. The book ...
1
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1answer
30 views

Is there an upper bound on Bell numbers?

For some reason my intuition is that $n^n$ might be an upper bound for Bell numbers, but I can't find anything to confirm that. Sorry if this is a simple question! (it's been a while since my ...