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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17 views

Measure of Connectivity on a Chessboard

I'm programming a boardgame...game. The basic idea of it is there are two players (call them $X$ and $Y$) that are trying to trying to build a wall connecting the North and South, and East and West ...
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18 views

Multivariable recurrence: Solving $c(n,k) = c(n-1,k) + c(n-1,k-1) = \binom{n}{k}$ by algebraic methods.

Let $(c_{n,k})_{n,k=0}^{\infty}$ be defined by $c(0,0)=1$, $\:\:c(0,k)=0 \:\: \forall \: k > 0$ $$c(n,k) = c(n-1,k) + c(n-1,k-1) \:\:\: \forall \: n \geq 1$$ I can show that the ...
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2answers
28 views

How to count the number of x in a rows in a larger set.

For example, I have 4 in a row like so: xxxx I can see that it has 2 xxx in it and 3 xx. ...
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1answer
23 views

Proof by strong induction combinatorics problem

$1(1!) + 2(2!) + 3(3!) + \dots + n(n!) = (n+1)! - 1$ How do we prove this by strong induction? I know how to do it with weak induction, but how would strong induction work with this problem?
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2answers
27 views

Find $n$ such that the congruence $x^n\equiv 2\mod 13$ has a solution for $x$.

Find $n$ such that the congruence $x^n\equiv 2\mod 13$ has a solution for $x$. I am not getting any idea how to start this problem. Please give some hits
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1answer
15 views

Arrangements of crew in two sides of a boat - permutations and combinations

A boat crew consist of 8 men, 3 of whom can row only on one side and 2 only on the other. The number of ways in which the crew can be arranged is This is a problem my math teacher has given ...
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3answers
38 views

Game is winnable if and only if $n \neq k$

Integers $n$ and $k$ are given, with $n \ge k \ge 2$. You play the following game against an evil wizard. The wizard has $2n$ cards; for each $i = 1, \ldots, n$, there are two cards labelled $i$. ...
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0answers
27 views

$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$ integer for $k \in \mathbb{N}$

How do I see that for any positive integer $k$,$$(k^2)! \cdot \prod_{j = 0}^{k = 1} {{j!}\over{(j + k)!}}$$is an integer?
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0answers
27 views

proof of number of sub arrays of an array of size $N$ using combinatorics

What is the proof that number of sub arrays of an array of size $N$ is $$\frac{N(N+1)}{2}$$
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0answers
15 views

Can you identify this stochastic process?

So I run into this problem the other day and I cannot even think of the keywords I need to use to look it up. For the discrete random variable $X$ we have: $P_{\Delta X(t)} = F\big(X(t-1), ...
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1answer
33 views

How do I find all n values for which the equation $\phi (n) = 8$ holds? [duplicate]

I've heard all kinds of different ways to solve this problem, yet haven't been able to apply them specifically to the number 8 (Worked fine for 6 for example). I'd love to see a well-explained ...
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2answers
525 views

Number of ways of visiting N places

A tourist wants to visit $N$ cities, each numbered from $1$ to $N$, but he wants to visit them in a weird order. A weird order is such in which no city numbered $i$ is the $i$-th to visit in his ...
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2answers
155 views

How many integers from 43523 to 93107 contain at least one digit 7

How many integers from $43523$ to $93107$ contain the digit $7$ at least once? I know that if we had $43000$ to $93000$, we would subtract integers that do not contain digit $7$ from the total ...
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0answers
25 views

Number of ways to get from a point to another one in the plane

I was trying to solve the following problem related to "counting cases": Consider the point $(0,0)$ in the plane and another point $(m,n)$ with $m,n>0$ integers. Suppose you want to get from the ...
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0answers
21 views

Real world uses or interesting facts about/for Associahedron or Permutohedron

I'm doing a small research project into these but their Wiki page and other pages I've looked at just detail what they are, and their properties. Does anyone know of any real world applications or ...
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0answers
27 views

The Divisors of $s(2s+1)$ and Primes $n$, $4n+1$, and $6n+1$

This question is somewhat related to this one. Most of this is by way of a computer search: claim: If $s$ is any positive integer I write $\varphi_{s} =s(2s+1)$. Let $\tau$ be the divisor ...
2
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1answer
19 views

For each of the following restrictions, find the smallest size n for strings over $\{a, b, c\}$ that can be used as codes for $27$ people.

For each of the following restrictions, find the smallest size $n$ for strings over $\{a, b, c\}$ that can be used as codes for $27$ people. a. There are $k$ $a$’s, $l$ $b$’s, and $m$ $c$’s and $k + ...
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3answers
64 views

How many distinct ways can the number be written as product of $3$ factors?

How many distinct ways can the number $126$ be written as a product of $3$ positive integer factors? I found that the prime factors are $126=2\times3\times3\times7$. But how to get number of ...
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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
14 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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27 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 ...
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0answers
10 views

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

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
37 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 ...
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1answer
21 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
29 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
23 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.
26
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6answers
343 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
36 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
21 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
39 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 $x+y\leq2k-1$. ...
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1answer
27 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
46 views

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

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
59 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
41 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 ...
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1answer
48 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
19 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 ...
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0answers
22 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!
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1answer
34 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
47 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?
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1answer
43 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
35 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
40 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 ...
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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
23 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
34 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 ...
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0answers
16 views

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 ...