Questions tagged [combinatorics]

For questions about the study of finite or countable discrete structures, especially 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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On some special 5-tuples in projective space $PG(3,2)$

Projective space $PG(3,2)$ has nice 5-tuples of points like $(1,0,0,0)$, $(0,1,0,0)$, $(0,0,1,0)$, $(0,0,0,1)$, $(1,1,1,1).$ This 5-tuple is "nice" because these points with their pairwise sums cover ...
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2answers
23 views

extracting a coefficient from formal power series multiplication

The Questions is Compute the value of $[x^n] \frac{(1+x)^n}{1-x} $ The solution is as follows: $[x^n] \frac{(1+x)^n}{1-x} $ = $[x^n](1+x)^n(1 + x + x^2 + x^3 + . . . ) $ = $\sum_{k=0}^n [x^k](1+x)^...
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1answer
21 views

Classification of simplicial complexes without boundary

I want to know how to construct all two dimensional simplicial complexes without boundary. The only examples of complexes without boundary I am aware of are manifolds and pinched surfaces. A pinched ...
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2answers
47 views

How many bit strings of length $20$ have exactly two 1’s and do not contain 11 as a substring? [on hold]

I'm not sure how to solve this. I know there are $2^{20}$ strings.
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2answers
47 views

Formal Power Series multiplication [on hold]

Compute the value of $[x^n] x^k(1-x)^{-k} $ For some reason I don't understand formal power series multiplication. Am i suppose to use the property of A(x)B(x) = $\sum_{i=0}^∞ a_i x^i$ * $\sum_{j=0}^...
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1answer
41 views

Number of angarams of BORBOTTIO in which same letters are close together

I had the following problem: We have the word BORBOTTIO. Find: All the anagrams All the anagrams that starts with BB All the anagrams where the same letters are close together. The first two ...
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1answer
28 views

expectation of number of hands shaken at round table

There are $x$ people from country A and $y$ people from country B. They sit around a table and shake hands with people on their left and right, but they only shake hands if they're from the same ...
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1answer
47 views

Even coin tosses, 2 players

The game consists of a sequence of independent plays for each of the two players. Each round I have probability $p$ of making a point and my opponent has probability $1-p$ of making a point. (so each ...
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0answers
41 views

A conjecture on tiling

Here's the problem: An $L-tile$ is the one which looks like this, and covers $3$ square units: A A A Now we will define a term called an $L_k-good$ rectangles where $k$ is a nonnegative ...
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16 views

Minimum difference between two subsets

This problem feels very basic, however I could neither google it nor solve it satisfactorily: Given a set of positive numbers $X \subset \mathbb{R^+}$, find the minimal nonzero difference between two ...
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18 views

Why is the “q-Pochhammer Symbol” referred to as such, despite not being a q-analog of the Pochhammer symbol?

The q-Pochhammer symbol (for $k>0$): $(a;q)_k = \prod\limits_{j=0}^{k-1}\left(1-aq^j\right)$ The Pochhammer symbol / rising factorial: $(a)_k = \prod\limits_{j=0}^{k-1}(a+j)$ The falling ...
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1answer
24 views

Finding the general recursive formula to divide the pot fairly

In a game where each turn is made up of the roll of a die, player E gets one point when the die is even, and player O gets one point when the die is odd. The first player to accumulate 7 points wins ...
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1answer
28 views

Finding a good bound for a quadratic and exponential expression of integers

Suppose you have two collections of $k$ positive integers, $a_{1},\ldots,a_{k} \in \mathbb{N}$ and $b_{1},\ldots,b_{k} \in \mathbb{N}$. Suppose you have bounds on the sum $\sum_{i=1}^{k} a_{i} \leq A$...
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82 views

Equality like Pascal triangle

I have noticed the following is true. Let's denote the equation below as (1) $$\sum_{k=1}^{d+1}(-1)^{d+1-k}\frac{1}{(d+1)(k-1)!(d+1-k)!}\prod_{i=1}^{d+1}\Big(\frac{q}{h}+(k-i)\Big)\prod_{j=k}^{d+1}\...
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2answers
91 views

How many different 7-digit numbers there are with exactly two “8” and three “4”?

How many different 7-digit numbers there are with exactly two "8" and three "4"? It is worth noticing that a number can not start with "zero", so: 0044488 is not a possible 7-digit number. This is a ...
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2answers
69 views

If you shuffle a deck of cards and then someone then shuffles it again is the deck more shuffled?

So we are having a discussion with my friends. One is saying that if he shuffles a deck of cards x amount of times and then somebody else shuffles it again, the deck after the second shuffle is more ...
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1answer
60 views

Win with positive probability implies there exists a strategy to win for sure?

Assume there are two players A and B who play a game on a finite game board (you can imagine it is some chess game or the position game. The key point is the total number of feasible sequences of ...
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1answer
18 views

What algorithms are there to partition graphs into bounded-diameter sets?

I'm working on a project in which I have a large graph, and I want to break it into clusters of nodes. Connected components turned out to be too loose a restriction, so I want to impose a diameter ...
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12 views

Recurrence Relation for the number of binary trees of order n [duplicate]

Looking to make a recurrence relation for the number of different binary trees of order n. Some initial conditions $a_0 = 1, a_1 = 1, a_2 = 2, a_3 = 5$ Say that I have a tree of order n, I have my ...
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1answer
48 views

Pigeonhole Problem: Prove that a subset's sum is divisible by 10

Given a sequence of $10$ integers, show that there is a subset of consecutive integers whose sum is divisible by $10$ Suppose I have subsets $$\{a_1\}$$ $$\{a_1,a_2\}$$ $$\vdots$$ $$\{a_1,a_2,a_3,...
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69 views

Smallest number not expressible using all of the first $n$ powers of $2$ (once each), with $+$, $-$, $\times$, $\div$, and parentheses?

Motivation Solution to this problem is a lower bound for a more general problem. Problem Given first $n$ powers of two: $1,2,4,8,16,\dots,2^{n-1}$ that all need to be used exactly once per number ...
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1answer
23 views

How many ID numbers exist? - check my solution

"An ID number is created as follows: The first part must consist of at least one letter and at most of four letters (the alphabet is assumed to have 26 letters). The second part must consist of at ...
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0answers
16 views

Prove that the total number of ingredients in all $n$ divisions is equal $\sum _{k=0}^{n-1} P(k)r(n-k)$

Let $r(m)=\sum _{d|m} 1$ which is number of positive dividers $m$ and $P(a)$ which is number of possible divisions of $a$. Prove that the total number of ingredients in all $n$ divisions is equal $\...
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1answer
14 views

Different soccer pairings in quarter finals - check my solution

"How many different pairings are possible in the quarter finals of a world cup where 32 teams participate in?" Is my solution correct? Let $\Omega:=\{\{\omega_1, \omega_2\}\times \{\omega_3, \...
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1answer
44 views

Counting the number of ways to divide into teams - complicated

$n$ students are standing in a row. Teacher must divide them into smaller teams - it could be one team or more - (team must consist of students standing next to each other in a row) and choose in ...
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4answers
33 views

Combinatorics problem: How many ways this row can be filled?

A row measuring $N$ units in length has $M$ red blocks with a length of one unit placed on it, such that any two red blocks are separated by at least one grey square. How many ways are there to ...
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0answers
50 views

How many conjugacy classes of subgroups of order $p$ does $\operatorname{GL}_{3}(\Bbb Z / p\Bbb Z)$ have?

It is well known that any two subgroups of order $p$ in $\operatorname{GL}_{2}( \mathbb{Z}/p\mathbb{Z})$ are conjugate. Then there is one possible semidirect product of the form $$(\mathbb{Z}/p\mathbb{...
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35 views

Marbles of different colors in a row.

We have 10 red marbles, 6 green and 5 blue (marbles of the same color are identical) and want to arrange them in a row in such a way that there are not any consecutive green marbles. In how many ways ...
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63 views

Winning strategies for games on the natural numbers

Define $$F=\{(l_n, k_n)_{n=1}^t: t,l_n, k_n \in \mathbb{N}, l_1<\ldots <l_t, m_1<\ldots <m_t\}.$$ Suppose that I have a collection $G\subset F$, which is a set of ''good'' sequences. ...
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2answers
186 views

Why does a bijection from a set to itself deserve the name “Permutation”?

Sorry for the long text; this is a nebulous question that has always been in the back of my mind, and I've had trouble putting into a short form. "Natural" Definition If someone on the street hears ...
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1answer
35 views

Compositions of a large integer

Let's say $n$ is the number of integers and $t$ is their sum and we want to find all the combinations of number $t$, such that there are $n$ components. See https://en.wikipedia.org/wiki/Composition_(...
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29 views

Combinatorial proof of a binomial identiity [duplicate]

Could anyone advise me how to prove the following using combinatorial proof? $$r\left(\begin{array}{c}n\\ r\end{array}\right)= n\left(\begin{array}{c}n-1\\ r-1\end{array}\right), r \geq 1$$ Hints ...
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1answer
32 views

Explicit formulas for Stirling numbers of the first kind?

I am solving this problem: What are the coefficients of particular powers of $x$ in polynomial $x*(x-1)*(x-2)*...*(x-k+1)$? I know, how to start, how to get a recurrence relation, but I do not ...
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2answers
37 views

Number of integers less than $20000$ that contain digits $4$ or $7$ or both

What's the best way of finding the number of positive integers less than $20000$ that contain the digits $4$ or $7$ or both?
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2answers
103 views

Distinct balls into distinct boxes

We have $n$ distinct balls ($n>7$) and want to randomly (and independently) distribute them into $N$ distinct boxes ($N>n$) which are placed one next to the other. a) What is the ...
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1answer
58 views

What is the number of possibilities to choose $~80~$ numbers out of the set $~\{10,11,\cdots,99\}~$ with repetition and no order significant

What is the number of possibilities to choose 80 numbers out of the set $~\{10,11,\cdots,99\}~$ with repetition and no order significant. In which if an element that divides by $10$ with no Remain of ...
4
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1answer
42 views

Given $N$ different elements, how many different Possibility there are to Push and Pop all the elements to\from stack

Given $N$ different elements, how many different possibilities there are to Push and Pop all the elements to\from a stack? In the beginning, the stack is empty and every element can be pushed to the ...
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2answers
42 views

Throwing a die three times - check my proof

"A teacher rolls a 6-sided die (numbers from 1 to 6) three times. The lowest number will be the grade of the student. Calculate the probability of each grade." My approach is: Firstly we determine ...
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2answers
49 views

Probability & Combination. N pairs of shoes in closet.

7 pairs of shoes (i.e., 14 shoes) are in a closet. 4 persons whose shoes are among the 7 pairs randomly selected 2 shoes each from the closet. What is the probability that exactly 2 persons selected ...
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3answers
69 views

Discrete Mathematics - Combinatorics proof

$\\$ I need to prove $\displaystyle\sum\limits_{k=0}^n\left(\frac{ 1^k+(-1)^k}{2}\right)\cdot\left(\begin{array}{c}n\\ k\end{array}\right)4^k=\frac{5^n+(-3)^n}{2}$ my Attempt below $\displaystyle\...
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0answers
50 views

This question is based on basic counting. [duplicate]

How many 10- digit numbers containing digits 1,2&3 can be formed such that the first & the last digits are same and no two consecutive digits are same?
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29 views

Number of 3 colorings on grid with constraints [duplicate]

I am trying to figure out a way to do the following: Given a $3 \times n$ grid, ($n \geq 2$) choose coloring by using $3$ colors (say, RGB), such that any given column cannot have all grids of same ...
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1answer
76 views

Equation involving sum of binomial coefficients.

Solve for $x$ if $$\sum_{i=0}^{16} {16 \choose i} 5^i = x^8$$ Not sure what to do here. Should I somehow use the binomial theorem to manipulate this to solve for $x$, or is there another approach ...
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0answers
33 views

selecting numbers from the group [on hold]

What is the number of options for selecting 80 numbers from the group {10.11 .... 99} with repetitions and no matter how orderly, each number that divides by 10 without residual is taken at least ...
2
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0answers
15 views

Bijective mapping to fixed image set

I need to reduce the amount of stored data in an Ethereum Smart Contract. Currently I am storing $n < 2^8$ strings in a mapping (i.e. a map data structure) and each string is mapped into $[0, n)$; ...
4
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2answers
32 views

How to get the bar and star formula?

I have this statement: Prove the bar and star formula for positive integers $\binom{n-1}{k-1}$ My attempt was: Imagine I have a certain equation: $x_1 + x_2 + ...+x_k=n$ Now, i have: $\...
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2answers
43 views

How many combinations of three dice given as a product an even number

I want to know how many combinations of three dice given as a product an even number. If these dice are different from each other the answer would be $6^3- 3^3$. What about if these dice are equal to ...
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0answers
24 views

Minimising the intersection of a collection of subsets of a set

This question specifically refers to a previous discussion from this website, namely, minimizing the intersection of three sets. As per that discussion, let us consider a finite set N, and a finite ...
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1answer
33 views

Limitations on the difference of coprime composites less than $n^2$ for ascertaining primality?

Let $q$ be a prime where $n < q < n^2$. Let $p_1, p_2, \ldots, p_k$ be all the primes $\leq n$. Is it true that for any $q$, you can find two numbers $a,b$ where $a+b=q$ or $a-b=q$, such that $...
4
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0answers
22 views

Asymptotic Gilbert-Varshamov Bound Using Hilbert's Entropy Formula

I am reading Walker's book Codes and Curves and am having trouble proving this Lemma regarding the Asymptotic Gilbert-Varshamov bound. Suppose that $q$ is a prime power and we define \begin{align*} ...