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.

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3answers
45 views

The number of choices of 3 kinds of crust and up to 6 distinct toppings

David has a pizza shop. There are 3 kinds of crust and 6 different toppings he can chose from. If customers can have as many toppings as they'd like but may not order double of one topping, how ...
0
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1answer
19 views

For a solution of linear recurrence relation, $\lim_{n\to\infty}a_n^{1/n}$ is a zero of a related polynomial

On page 134 of J.H. van Lint's book A Course in Combinatorics, it says from $a_n=5a_{n-1}-7a_{n-2}+4a_{n-3}$ $(n\ge5)$, we find that $\lim_{n\to\infty}a_n^{1/n}=\theta$, where $\theta$ is the ...
1
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1answer
22 views

2x2 grid game problem

A friend of mine is attempting to make a webpage that has a game for a 2x2 grid that is similar to the old North, South, East, West game. I cannot for the life of me figure this out. Essentially, ...
0
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0answers
37 views

The coefficients of $\frac{1}{cos(x)}$ are even

Let's consider $G(z)=\dfrac{1}{cos(z)}$ as an exponential generating function of the Euler numbers' sequence. How to prove that all $a_{i}$ in the expansion of$\dfrac{1}{cos(z)}=\sum_{k=0}^{ ...
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1answer
19 views

Rewrite the sum of the products by interpretation

By interpreting what the following sum is counting and then counting the same object in a different way, rewrite the following sum as a product of two terms (without any sum): $\sum\limits_{k=m}^n$ ...
0
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0answers
17 views

How to find the number of words of length $h$ in a subsets $A$?

Let $L=\{0,1\}^*$ (the set of binary words on $0$ and $1$), Given a tuple of words $(w_1,w_2,\cdots,w_n)\in L^n$ and a function $\sigma:[1,n]\to [1,n]$ define the following set: ...
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2answers
21 views

Permutations; group of 5 boys, 10 girls. What's the probability the person the 4th position is a boy?

Problem description: A group of 5 boys and 10 girls is lined up in random order -- that is, each of the 15! permutations is assumed to be equally likely. What is the probability that the person in ...
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1answer
24 views

Some men and women are randomly assigned seats at a round table and no two persons of the same sex are seated next to each other. Probability of this?

Four women and four men are assigned seats at random at a round table. what is the probability that no two persons of the same sex will be sitting next to each other?
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0answers
29 views

Riddle: Assigning Students into Groups

Suppose you had a classroom with 25 students. You want to assign 6 homework assignments over the course of the term and for each of these assignments students will work in groups of 5. But you want to ...
1
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0answers
17 views

Counting problem of combinations of symmetric matrix.

Given, a symmetric $n*n$ matrix $G$ with 0,1 entries. Each row of has same number of 1. $G$ is arranged in a certain order using a rule. The rule is described below- $G$ is partitioned in to two sub ...
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1answer
28 views

no. of disordered pairs of disjoint subsets

I found this question in a book. The same question has been asked before, but I want a more generalised and rigorous, so to speak, answer. The question reads- " Consider the set $S= \{1,2,3,4\}.$ ...
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2answers
40 views

Consider all the permutations of the word “ENDEANOEL”

Consider all the permutations of the word "ENDEANOEL" : 1)What is the number of permutations containing the word "ENDEA" ? I can't understand how to approach this problem!! 2)Number of permutations ...
0
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1answer
16 views

Reverse permutation, number of inversions, descents, major index

If $w=a_1a_2...a_n \in S_n $, then let $w^r=a_n....a_2a_1$, the reverse of $w$. Express inv($w^r$), des($w^r$) and maj($w^r$) in terms inv($w$), des($w$), maj($w$), respectively. It from Stanley's ...
2
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1answer
26 views

How do people find the number of ways you can put together a rubiks cube?

Just curious. How do people actually find the number of ways you can put together a rubiks cube? How do you find the number of choices? Do you use the same permutation formula? Insight would be ...
3
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3answers
100 views

Is this permutations or combinations?

I am a bit confused. When we use the multiplicative principle are we finding the number of permutations or combinations. An example of using this principle is where I have $5$ shirts $3$ pairs of ...
0
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1answer
35 views

Combination formula?

I know there is a formula for finding the different combinations when you are dividing them in groups: $$\binom{n}{r} = \frac{n!}{(n-r)!\, r!}$$ However, what if you just want to find the number of ...
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0answers
19 views

Other than the icosahedron in which each vertex has degree 5, is there any triangulation of the sphere that meets the following three conditions?

Every vertex has degree > 3. There is no separating triangle (a triangle with vertices of the graph both inside and outside the triangle). Every vertex-coloring using exactly four colors consists of ...
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0answers
45 views

some amazing properties of combinatorial numbers [on hold]

I want to prove $$ C_{2^{i+1}-k-1}^k=\frac{(2^{i+1}-k-1)(2^{i+1}-k-2)\cdots(2^{i+1}-k-(k-1))(2^{i+1}-2k)}{k(k-1)\cdots 2\cdot 1} $$ is even, for all $k=1,2,3,\cdots, 2^i-1$. Here $i\geq 1$. How to ...
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2answers
22 views

How many ways are there to arrange three of the letters chosen from the set ABCDE? [on hold]

Please show your work. I've been looking at this problem for over an hour now and havn't been able to solve it. Thank you!
0
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1answer
37 views

Number of paths in a graph with infinite nodes

Does a graph with infinite nodes that is not fully connected have a countably infinite or a uncountably infinite number of paths originating from a single node? We are only concerned with paths that ...
0
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2answers
19 views

Prove that if $k\mid n$ then $p(A_k)={1\over k}$

Let $n$ be a natural number, $n=p_1^{a_1}\cdot...\cdotp_m^{a_m}$. Let us randomly choose a number between 1 and $n$ with a uniform, equal chance. Let us denote the event "The number chosen is ...
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1answer
34 views

Need help in solving [on hold]

A group of $60$ children attend an after school club. Of these, $35$ children play football and $29$ play hockey. Three children do not play either football or hockey. Find the number of children ...
0
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1answer
30 views

Generating function of derangements

I am pretty new to the topic of generating functions and I would appreciate if someone could help me out with this problem I have. In the lecture we have proven the following generating function for ...
1
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1answer
44 views

What is the probability that you get $i$ on the $i^{th}$ trial?

What is the probability that you get $i$ on the $i^{th}$ trial? Match = Get result $i$ on $i^{th}$ trial. What is the probability of $M = 0,1,2,...,6$ matches when: Note: I'm not asking you to do ...
5
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3answers
98 views

Find the coefficient of $x^{30}$.

Find the coefficient of $x^{30}$ in the given polynomial $$ \left(1+x+x^2+x^3+x^4+x^5+x^6+x^7+x^8+x^9+x^{10}+x^{11}+x^{12}\right)^5 $$ I don't know how to solve problems with such high degree.
1
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2answers
40 views

Combinatorics Recurrence relation

Let $h_n$ be a number sequence where $h_n = 3h_{n-1} - 2h_{n-2}$ with $h_0 = 0$ and $h_1 = 1$. Compute the ordinary generating function of $h_n$ and then using the generating function compute a ...
2
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2answers
18 views

Probability of Two Suits within Three cards, within 4 cards

I am trying to calculate what is the probability of the 3 random cards of 52-pack containing at least two of the same suit. I am also trying to do the same for the four card variant (so, the ...
0
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4answers
51 views

Combinatorial Proof of an Instance of the Binomial Theorem

Give a combinatorial proof of the following instance of the binomial theorem. For any positive integer $k$, $(k + 1)^{n}$ = $\sum\limits_{i=0}^{n}$ ${n}\choose{i}$$k^{i}$. I have looked at this for ...
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0answers
27 views

Combinatorics: Password consisting of 13 characters. Must contain at least one odd digit, and at most two even digits. How many passwords?

I'm really trying here. I just need help where to go next. Each character is one of the 10 digits 0, 1, 2, ... , 9 What I have so far is that there are 10^13 possible passwords. I'd have to subtract ...
1
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0answers
29 views

Proving Crapo's Lemma

Let $L$ be a finite lattice with least and greatest elements $0, 1$, respectively, and let $X\subseteq L$. Let $n_k$ be the number of $k$-element subsets of $X$ with join $1$ and meet $0$. I want to ...
0
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0answers
16 views

Combinatorics Question for generating fuctions [on hold]

Any tips/helps would be greatly appreciated! Let h_n be a number sequence where h_n = 3h_(n-1) - 2h_(n-2) with h_0 = 0 and h_1 = 1. Compute the ordinary generating function of h_n, and then compute a ...
0
votes
1answer
18 views

All subsets of nonnegative integers such that $a+2b = n$ has one solution for every positive integer n

My friend tackled this problem awhile ago and gave it to me recently. To reiterate, I am trying to find all subsets $S$ of the nonnegative integers such that the equation $a+2b = n$, where $a$ and $b$ ...
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1answer
29 views

Combinatorics Generating Functions [on hold]

Any tips/comments would be greatly appreciated! Compute the generating function of the number sequence $h_n = (-2)^n n^2$ where $n\geq 0$.
0
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0answers
6 views

Finding a permutation class that has a growth rate greater than 1 and less than 0?

In a permutation class, there is an upper growth rate such that $gr(C)=\limsup_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$ and a lower growth rate such that $\liminf_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$. ...
1
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1answer
43 views

Binomial Coefficients and Function Composition

I found a paper that stated the following without proof. I tried to prove it on my own, but so far to no avail. Define $\varphi^{+}: \mathbb{N}^2 \to \mathbb{N}$ by $\varphi^{+}(i,j) = i + j$. ...
2
votes
1answer
50 views

Combinatorics Question VS CS solution!

I was wondering for some conceptual understanding to a question of this form: In how many ways may we choose three distinct integers from [1, 2, ..., 80] so that one of them is the average of the ...
0
votes
1answer
17 views

Sets of non-complements elements in a lattice.

Let $L$ be a finite lattice with a least element $0$ and a greatest element $1$, where $0\neq 1$. Fix a $t\in L$, and let $X$ be the set of non-complements of $t$, i.e., the set of all $x$ such that ...
2
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0answers
34 views

Coupon collector variation (with deleterious coupons and tolerance)

Imagine the standard coupon collector's problem, with n coupons to be collected. However, the sample space also contains T bad coupons. Specifically, if during the collection, I collect more than t (t ...
1
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3answers
54 views

Prove that $x^2 - 2013^2 \le y \le 2013^2 - x^2$ has an odd number of solutions

$x$ and $y$ are integers. $N$ is the number of solutions $(x, y)$ of this inequation $x^2 - 2013^2 \le y \le 2013^2 - x^2$. Prove that N is odd.
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2answers
56 views

How many sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels?

How many sequences of length 6 are formed from the 26 letters without repetition where the first or last letter (possibly both) must not be vowels? I am so lost and confused, but here's my approach: ...
0
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0answers
13 views

Mobius function on posets

Let $A= \lbrace 1^{a_1},2^{a_2},...,n^{a_n} \rbrace $ and $B=\lbrace 1^{b_1},2^{b_2},...,n^{b_n} \rbrace $ multisets for which : $A\leq _P B \Leftrightarrow $ for all $i=1,2,...,n $ is $a_i\leq b_i$. ...
1
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1answer
35 views

What is number of p-point subgraphs in n-point graph with average t connections?

I don't know graph theory, but I want to study this specific question for a while. I have no idea if this is a well known and studied question or not. I found it very difficult, and I don't know where ...
0
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3answers
39 views

Product Rule Notation Meaning

Let $S_1,...,S_t$ be finite sets and let $S=S_1 \times ... \times S_t$. The product rule states that $$|S|= _{i=1}^t S_i$$ There is supposed to be some big pi symbol in between the limits which i ...
2
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0answers
16 views

Optimality of lower bounds for Max-cut on specific graphs

The Max-Cut problem asks to find a subset $S$ of the vertices of a graph (with $m$ edges) such that the number of edges from $S$ to it's complement is as large as possible. The size $|M|$ of a max cut ...
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0answers
32 views

Permutation Of 2 Groups

The Following question is from "A FIRST COURSE IN PROBABILITY" of Sheldon Ross A class in probability theory consists of 6 men and 4 women. An examination is given, and the students are ...
1
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1answer
25 views

Difference : subsequences and substrings [on hold]

What are the differences between subsequences and substrings?
1
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2answers
248 views

Number of ways to distribute 55 red balls and 3 green balls

Fifty-five identical red balls and three identical green balls are to be distributed among seven children. Each child must get at least five balls. In how many ways can this be done? What I have so ...
0
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0answers
6 views

On the interval minor extremal function of a j × k matrice.

I was going through papers by Marcus/Tardos and Fox and I have this small doubt. If L is a j×k matrix which has every entry equal to 1, what is the interval minor extremal function of L? Can someone ...
1
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1answer
22 views

Find all possible two-way associations/relations between four numbers

Given four numbers {1,2,3,4}, how to find all possible two-way associations/relations between them? I calculate them manually as in below (50 in total) but I would like to know whether a mathematical ...
1
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1answer
61 views

Simplifying Sum

How would one show that $$ \sum_{i=0}^n\binom{n}{i}(-1)^i\frac{1}{m+i+1}=\frac{n!m!}{(n+m+1)!} ? $$ Any hint would be appreciated. Note: I tried to recognize some known formula, but since I don't ...