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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7 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 ...
-4
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0answers
37 views

some amazing properties of combinatorial numbers

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 ...
2
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1answer
100 views

Probability of selecting the winning numbers in a lottery

I've been studying combinatorics for a while. I've solved a problem but I'm not sure if I'm right. I'll just copy-paste the problem here. In a lottery, six distinct numbers are selected at random ...
-2
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2answers
15 views

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

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!
3
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5answers
168 views

Curious Binomial Coefficient Identity

Consider the following set of identities: ${m+1\choose 1}={m\choose 1}+1$, ${m+1\choose 2}=2\binom m 2 - {m-1\choose 2}+1$, ${m+1\choose 3}=3\binom m3-3{m-1\choose 3}+{m-2\choose 3}+1$, ... This set ...
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1answer
33 views

Why do (the ranges of) these sequences intersect?

Let $\{(a_n,b_n)\}$, ($1\le n\le N$) be a finite sequence and $\{(s_n,t_n)\}$ ($n\ge 1$) be an infinite sequence, both in $(\{0\}\cup \mathbb{Z}^{+})^2$. We have $a_1=0$ and $b_N=0$. Also, either ...
-1
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1answer
31 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 ...
-4
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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 $n\geq 0$.
3
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1answer
97 views

Complex Analysis proof of multinomial expression

I've recently come across the following identity $$ \displaystyle \sum_{k = 0}^n {n \choose k}^2= {2n \choose n} $$ A nice complex analysis proof (by Felix Marin, here) follows as: ...
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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 ...
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4answers
46 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
41 views
+50

Product of Stirling Numbers of the first kind

I have been messing around with coefficients of various polynomials and was wondering if there was a way to reduce the following stuff. Let polynomial, ...
2
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0answers
31 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 ...
0
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1answer
22 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 ...
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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 ...
-6
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1answer
33 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 ...
1
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1answer
43 views

How many solutions of equation

How many solutions of equation $x_1+x_2+x_3+x_4=n$ in $N_0$ such that $x_1\leq x_2\leq x_3 \leq x_4$? I found solutions of $x_1+x_2+x_3=n$ in $N_0$ , $x_1\leq x_2\leq x_3 $ in the following way : ...
1
vote
1answer
23 views

Difference : subsequences and substrings [on hold]

What are the differences between subsequences and substrings?
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1answer
53 views

Proving $\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$

Prove the identity: $\displaystyle\binom {n-1}{r-1}=\sum_{k=0}^r(-1)^k\binom r k \binom{n+r-k-1}{r-k-1}$ It looks a bit similar to the "no gets their own hat back" problem or inclusion exclusion ...
1
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1answer
43 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
95 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.
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0answers
28 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 $\varphi^{+}(i,j) = i + j$. ...
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2answers
691 views

How many ways I can place some, possibly all, five distinct balls into three distinguishable bins?

I want to know how many ways I can place some, possibly all of five balls each a distinct color into three distinguishable bins. Each bin must have at least one ball and I do not need to use all of ...
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2answers
37 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
17 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 ...
2
votes
1answer
26 views

Number of ways to select AT LEAST one item from 12 different items. The items are divided into two sets, each of size 6

The answer says 4095. Now, as per my understanding : $4095 = 2^{12} - 1$ == Ways of getting a non-null subset out of 12 elems That would make sense, but where does the "divided into two sets, each ...
1
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2answers
54 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
votes
1answer
16 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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0answers
25 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 ...
0
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0answers
27 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
13 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 ...
1
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0answers
158 views

What is the nature and location of maxima of expected log-utility?

Is is possible to describe, and locate efficiently, the maxima of the function below in the parameters $\mathbf{x}$ $$\sum_{i} p_i \log( N + \sum_j x_j[B_j +(A_j-B_j)\delta_{ij} + min(A_j,B_j) ]) ...
0
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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 ...
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 $\liminf_{n\rightarrow \infty}=\sqrt[n]{|C_n|}$. ...
2
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0answers
24 views

Find number of $r$-element subset of $S$ satisfying a property

Let $S= \{1,2,...,1990\}$. A $31$-element subset $A$ of $S$ is said to be good if the sum of all the elements of $A$ is divisible by $5$. Find the number of $31$-element subsets of $S$ which are good. ...
2
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1answer
46 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 ...
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.
2
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1answer
57 views

Number of orbits of $G$ acting on $X$

This question comes from Algebraic Combinatorics: Walks , Trees, Tableaux, and More by Richard P. Stanley. It is written as follows: "Let $X$ be a finite set, and let $G$ be a subgroup of the ...
2
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6answers
5k views

If there are 50 notes whose total value is 100 rupees but 2 rupee note should not be there in the count of those50 notes How many such notes can be?

If there are 50 notes whose total value is 100 rupees but 2 rupee note should not be there in the count of those 50 notes.How many such notes can be ? Notes available are $1$ Rupee $2$ Rupees ( but ...
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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 $a_i\leq b_i$. ...
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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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 ...
1
vote
1answer
21 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 ...
0
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1answer
40 views

Number of permutations of $A=\{1,2,3,\dots,n\}$ such that $|x_i-x_j|\ne|i-j|$ of every $i,j\in A$

Let $B$ be the permutation of $A=\{1,2,3,\dots,n\}$ such that $|x_i-x_j|\ne|i-j|$ of every $i,j\in A$ where $x_k$ is $k-th$ element of $B$. How many different $B$ exist? On first sight it doesn't ...
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0answers
31 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 ...
3
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3answers
95 views

Find the number of ways to form 15 teams out of 15 men and 15 women.

In how many ways can 15 teams be formed, each consisting of a man and a woman, from 15 men and 15 women. This looks like the same problem as finding the number of bijective functions from a set $A$ ...
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1answer
44 views

Cryptography Combinatorics question [on hold]

I 'invented' this encryption device - take a string, and start with the first character. Swap this character with the second with probability $50$%. Move to the (now) second character, and repeat ...
1
vote
2answers
246 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 ...
1
vote
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 ...
0
votes
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 ...