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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1answer
24 views

What is the probability Amy wins a lottery prize for correctly choosing 5, not six, numbers…

Here is the full question: What is the probability that Amy wins a lottery prize for correctly choosing 5, not six, numbers out of six integers chosen at random from the integers between 1 and 40 ...
2
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1answer
21 views

Half primes in the set

Let S be 30 element subset of {1,2,....2015} such that every pair of elements in S are relatively prime. Prove that at least half of the elements in S are prime numbers
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0answers
34 views

Combinatorial Nullstellenatz riddle

I've been unable to solve the last problem here: http://www.mit.edu/~evanchen/handouts/BMC_Combo_Null/BMC_Combo_Null.pdf Let $n ≥ 2$ be even and let $v_1, v_2, . . . , v_k ∈ \{±1\}^n$ be vectors of ...
0
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0answers
11 views

Number of nodes (or vertices) with degree at most average degree + some constant [closed]

I'm struggling with a problem of graph theory. In any graph I'm trying to compute how many nodes have degree at most average degree + 1 (or some constant independent of the graph). Obviously there ...
3
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2answers
42 views

What is probability that out of the first half on N objects, none will be matched with their own label?

The problem: We have N (even) objects ordered $o_1 ... o_N$ , each having their own label. The labels are reassigned to the objects randomly. What is the probability that that neither of the first ...
2
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1answer
22 views

Understanding derangement.

From the inclusion-exclusion principle we get that out of $N$ objects with one label each, there is a probability of $$\sum_{k=1}^N (-1)^{k+1}\frac{1}{k!}$$ that a random assignment of the $N$ labels ...
3
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1answer
30 views

How many straight lines can be made between 10 points such that 4 of them are colinear?

So i know how to get the answer. We just have to find $C(10,2)$ and subtract $C(4,2)$ and add 1. We are basically counting all the points between co-linear points as 1. So the question is why we are ...
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1answer
25 views

How many different arrangements of triangles that are either red or blue around a regular heptagon are possible?

I have the following problem I have an yellow heptagon (regular $7$ sided polygon) Against every side there is a triangle. The triangle is either red or blue. How many different arrangements of ...
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1answer
26 views

Number of non periodic strings

How many non-periodical strings of length N with letters from a to z exist? My only idea was something about prime factorization to find the amount of periodical strings of length N.
33
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6answers
587 views

You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life.

This is from the lecture notes in this course of discrete mathematics I am following. The professor is writing about how fast binomial coefficients grow. "So, suppose you had 2 minutes to save your ...
2
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1answer
37 views

How does the multiplication law creates order?

I have the following question : There are $2n$ students divided to couples to do homework. Using the multiply law we can choose the first couple then the second then the third couple and so on. The ...
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0answers
23 views

arranging $n$ objects of one kind and $m$ objects of other kind in a row

Why are there precisely $\binom{m+n}{n}$ ways of arranging $M$ objects of one kind and $N$ objects of other kind in a row?
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0answers
43 views

Confusion About “Stars and Bars” Method

Let's suppose I were trying to count the number of nonnegative integer solutions to the equation $x+y<2k$ for $x,y,k$ nonnegative integers. This is, of course, equivalent to solving $x+y\leq2k-1$. ...
0
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1answer
33 views

What book about algebraic combinatorics is it?

Recently I found a fragment of a book about algebraic combinatorics on the internet coincidentally. And I found it's really an excellent resource of learning polynomial method, about Combinatorial ...
3
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2answers
50 views

How many onto functions are there from a set with $5$ elements to a set with $3$ elements? [closed]

Consider functions from a set with $5$ elements to a set with $3$ elements. (a) How many functions are there? (b) How many are one-to-one? (c) How many are onto? a) Each element mapped to $3$ images. ...
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1answer
62 views

A sum of Stirling numbers of the second kind

Find a formula (either exact or asymptotic in $N$) for $S(N)$, where \begin{equation} S(N) = \sum_{n=N}^\infty \sum_{k=N}^n \sum_{j=0}^k \binom{k}{j} (-1)^{k-j} (1+j)^n \frac{t^n}{n!}. \end{equation} ...
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2answers
44 views

How many strings of six lowercase letters have at least one vowel?

The English alphabet has $21$ consonants and $5$ vowels. How many strings of six lowercase letters have at least one vowel? My attempt: I'm confused between using combinations and just ...
1
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1answer
51 views

proof by CP$ \binom{m}{1} S_{1}(n)+\binom{m}{2} S_{2}(n)+\binom{m}{3} S_{3}(n)+ \cdots +\binom{m}{m-1} S_{m-1}(n)=(n+1)^m-(n+1)$

I would appreciate if somebody could help me with the following problem: Q: How to Proof (by combinatorial proof) $$ \binom{m}{1} S_{1}(n)+\binom{m}{2} S_{2}(n)+\binom{m}{3} S_{3}(n)+ \cdots ...
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2answers
24 views

Two discrete r.v. problem, joint density

Problem A cook needs two cans of tomatoes to make a sauce. In his cupboard he has $6$ cans: $2$ cans of tomatoes, $3$ of peas and $1$ of beans. Suppose that the cans are without the labels, so he ...
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0answers
20 views

Basic probability problem with negative binomial distribution

John goes to the grocery. His mother sent him to buy $20$ peaches and requested him to be sure that the peaches were mature. Suppose the probability of a peach of being mature is $p$ and suppose that ...
0
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0answers
23 views

Find the number of n- digit ternary sequences with at least one instance of consecutive 0's.

I know how to do this problem with binary sequences but I have no idea how to start with ternary sequences. Any help would be great!
0
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1answer
37 views

A binomial-related inequality

For integer $m\geq 1$, show that: $$\sum_{|k|<\sqrt{m}}{2m \choose m+k}\geq 2^{2m-1}.$$ What I have tried: I tried binomial expansion of $2^{2m}$ but it was unsuccessful. Any other idea?
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2answers
49 views

how many ways are there to distribute 48 identical balloons to 7 children if each child gets at least one balloon

I understand how to get the generating function (g(x) = (e^x) - 1, I believe) but I am having trouble finding the coefficient. Any ideas?
1
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1answer
46 views

How many distinct patterns exist for a 5x5 grid by filling 3 colors?

Using 3 colors to fill in a $5\times5$ grid (you don't have to use all colors), then how many distinct patterns exist? The "distinct" means we have to consider the symmetry. Any effective approach is ...
2
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2answers
40 views

Counting the number of ways (variants)

I'm learning about combinatorics and wanted to see if I understand when to apply what methods when it comes to counting the number of ways to distribute x items. There are a lot of concepts I've ...
4
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3answers
42 views

A walk on the chessboard with conditions!

A 16 step path is to go from (-4,-4) to (4,4) with each step increasing in either the x-coordinate or the y-coordinate by 1. How many such paths stay outside or on the boundary of the square ...
1
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1answer
24 views

How many ways are there to arrange the letters of word $ALGEBRA$ such that the relative order of the vowels and consonants doesn't change?

I did this question this way :- there are 4 consonants in the words (LGBR) and there are 7 letters in the word. $therefore$ number of in which consonants can be arranged in relative order will be ...
2
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1answer
27 views

Why is d in A(n,d) not always equal to 1?

In Communication Theory, for $A(n,d)$ (=the size of a largest code of length $n$ and minimum distance at least $d$), why is $d$ not always equal to $1$? If min. distance $= d$, for any code of length ...
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0answers
33 views

PhD in Combinatorics (instead of Mathematics) [migrated]

In recent years I have become aware of a few PhD programs specifically in combinatorics and optimization. Most notably, Georgia Tech and Carnegie Mellon both have PhD programs in Algorithms, ...
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0answers
15 views

Combinations with up to m repetitions [duplicate]

I have a variation of the standard problem of combinations (order unimportant) with repetitions. The twist is that the number of repetitions is limited. If we take the ice cream flavor example from ...
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4answers
35 views

How many mixed double pairs can be made from 7 married couples provided that no husband and wife plays in a same set?

So for first man there can be 7 possible partners including his wife, for the next man there will be 6 possible partners and so on, $therefore$ for $7$ men and $7$ women, there will be $7!$ possible ...
2
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0answers
36 views
+50

Number of paths between two points in the first Quadrant.

[Extension of this] We can move in 4-directions and we need to reach $(0,b)$ from $(a,0)$ in exactly $n$ steps keeping in the first quadrant ($x\ge0$ and $y\ge0$) [$a,b\ge0$] Similar to previous ...
3
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1answer
77 views

Proving $\sum_{s \in S} \frac{1}{n}$ converges for $S = \{ s \in \mathbb{N} : s$ has no zeros on its decimal representation $\}$

Consider $S \subset \mathbb{N}$ as the set of numbers which do not have the algarism $0$ on its decimal representation. For instance: $$S=\{1,2, \dots, 9, 11, 12,\dots, 19, 21, 22, \dots\}$$ I want ...
1
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2answers
48 views

arithmetic mean of smallest numbers of all subsets of r elements formed out of (1,2,..n)

Consider all subsets of r elements of the set $\{1,2,3,......,n\}$ where $1 \leq r \leq n$. Each of these subsets has a smallest member. Let $F(n,r)$ denote the arithmetic mean of these smallest ...
4
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0answers
39 views

Find the sum of $\binom{2007}{0}+\binom{2007}{4}+…+\binom{2007}{2004}$ [duplicate]

Find the sum of $$S=\binom{2007}{0}+\binom{2007}{4}+\binom{2007}{8}+...+\binom{2007}{2004}$$ My work so far: $$(1+1)^n=2^n=\binom{n}{0}+\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n}$$ ...
1
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1answer
20 views

How do I create a minor of a $K_5$ or $K_{3,3}$ configuration from this $10$ vertex graph?

I have a graph with $10$ vertices, all of which are degree $3$: I am trying to show it is either planar or nonplanar, so I use the circle-chord method to create a circuit $abcdefghija$ (easy since ...
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1answer
16 views

There are 14 identical objects that will be placed into 3 boxes. In how many ways can this be done?

For this combination problem, I used the formula for combination (n + k - 1) choose (k - 1) to get the answer of (14 choose 2). Is this correct? If not, can someone explain what I did wrong?
3
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0answers
42 views

Probability problem of fishes in a lake

Exercise In order to estimate the number $N$ of fishes in a lake, a fisherman executes the following procedure: in the first step, he captures $n$ fishes and after marking them, he returns them to ...
2
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1answer
18 views

Understanding the proof of catalan numbers using lattice paths

I am trying to understand a proof to come up with the catalan numbers presented in the book "A course in combinatorics" by van Lint and Wilson. The authors say that by reflecting the part of the path ...
0
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1answer
27 views

Literature on generating functions for networks

Are you aware of any material the presents all (or most, or many) the properties and applications of generating functions in the context of graphs? For example I am aware of 'Generating ...
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1answer
15 views

Is this the correct way of drawing a combinatorial circuit based on the disjunctive normal form and logic table?

The logic table: $$\begin{array}{|c3:c|}\hline x & y & z & f(x,y,z) \\\hline 1 & 1 & 1 & 0 \\ 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ 1 & 0 & 0 & ...
0
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1answer
33 views

Can someone explain the particular solution for non homogeneous recurrence relations?

This is the recurrence relation: $a_n=5a_{n-1} - 6a_{n-2} + 4^n + 2n + 3$ for $n\geq2$ , $a_0 = 5, a_1 = 19.$ I get the general solution. $ c_n = C_12^n+C_23^n.$ The particular solution is in the ...
4
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3answers
86 views

Winning All Levels in a Game

There are $L$ levels in a game. In each turn of the game, you go through each level one by one and try to complete it. The goal is to complete all levels of the game. The probability of completing any ...
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2answers
29 views

Help resolving particular solution to recurrence relation?

$a_n=5a_{n-1} - 6_{n-2} + 4^n + 2n + 3$ for $n>=2$ , $a0 = 5, a1 = 19.$ I get the general solution $ c_n = C_12^n+C_23^n.$ For a particular solution in the form $pn = An + B + C4^n$; we have ...
0
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1answer
36 views

How many words of length n over the alphabet {a,b,c} such that the sub-word aa does not appear?

The question asks that it be solved as a recurrence relation, as in set up a recurrence relation then determine initial values to give a solution. However I am not really confident setting up ...
2
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3answers
39 views

Is there always $B\subseteq A$ with $f(B)=B$?

Let $f:A\rightarrow A$ be a function between finite sets. Is there always a non-empty subset $B\subseteq A$ with $f(B)=B$ ? I think there is, but I am not sure how to prove it.
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2answers
35 views

A question of permutations and combinations with six cards and six envelopes.

Six cards and six envelopes are numbered 1,2,3,4,5,6 and cards are to be placed in envelopes so that each envelope contains exactly one card and no card is placed in the envelope bearing the same ...
1
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2answers
22 views

How do I calculate the number of unique permutations in a list with repeated elements? [duplicate]

I know that I can get the number of permutations of items in a list without repetition using (n!) How would I calculate the number of unique permutations when a ...
0
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1answer
99 views

finding the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$ [closed]

What is the smallest number $n$ such that $n!=n(n+1)(n+2)(n+3)$? How will I solve this type of problems?
5
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3answers
88 views

Probability with n dice

I'm studying probability and am currently stuck on this question: Let's say we have n distinct dice, each of which is fair and 6-sided. If all of these dice are rolled, what is the probability that ...