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
25 views

Number of monotonic paths in a rectangular grid avoiding certain points

In a rectangular grid of size $m \times n$, the number of paths from $(0,0)$ to $(m,n)$ (without backtracking) is ${m+n \choose {n}} = \frac{(m+n)!}{(m!*n!)}$. Now if there are certain points in the ...
1
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1answer
36 views

How many distinct directed acyclic graphs are there?

Given $|V|=4$ and $|E|=3$, how many distinct directed acyclic graphs can be formed? Isomorphic graphs should be counted as one. There is one where three periphery nodes point to a central node. ...
1
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1answer
29 views

Catalan Sequence on a Circle

A Catalan sequence of length $2n$ is a sequence of $1$'s and $0$'s such that no initial segment of the sequence has more $0$'s than $1$'s. The number of such sequences is given by the Catalan number ...
4
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2answers
64 views

Let $S$ be a set consisting of all positive integers less than or equal to $100$.

Let $S$ be a set consisting of all positive integers less than or equal to $100$. Let $P$ be a subset of $S$ such that there do not exist two elements $x,y\in P$ such that $x=2y$. Find the maximum ...
0
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2answers
57 views

How many different chains in a Poset? [duplicate]

I found that problem and I could use some help. I have a partial order $(2^S,⊆)$ and |S| = n. How many different chains are there in that poset? If I had the Hasse diagram or knew the ...
0
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1answer
47 views

Work and efficiency puzzle

There are $2$ people $A$ and $B$. $A$ requires $a\;$ days to complete certain amount of work and $B$ requires $b\;$ days to complete the same amount of work. If $A$ begins the work a day before $B$ ...
1
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1answer
70 views

Exact Expected Value of Random Walk?

i just read in Noga Alon's Book That the exact expected value of a random walk is which was a question in putnam competition... Sn=X1+X2+...Xn Which Xi are independent uniform random in {-1,+1} ...
3
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1answer
58 views

n points permuted on a circle

Here is a combinatorics problem that bothers me a lot. I am looking forward to a quick reply. Thanks in advance. Here goes the problem. Initially there are $n$ points on a circle. We do permutation to ...
0
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1answer
42 views

Partial sums of periodic sequences

Let $a_i$,$b_i$ be two periodic real sequences with a period of $n$. For $k\leq n$, denote the $k$-length partial-sums starting at $j$ by $a[j:k],b[j:k]$, i.e: $$a[j:k] = \sum_{i=j}^{j+k-1}a_i\,\,\,\,\...
0
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0answers
33 views

How many passwords are possible if the characters may be used more than once? [closed]

I've solved the first two, the only information needed is the password must be 7 characters in length. What I think is for each slot a character may be repeated more than once in any spot. We only ...
0
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1answer
43 views

Probability book choosing questions

So I am doing homework and have the following question If 3 books are picked at random from a shelf containing 5 novels, 3 books of poems, and a dictionary. What is the probability that (a) the ...
0
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2answers
38 views

Expected value of colors picked from basket

I have a basket with 4 balls with different colors. What is the expected value of distinct colors I can see after picking 4 times from bin. I return the ball back after each try. I tried computing ...
3
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1answer
51 views

Combinatorial proof for a non obvious binomial identity

I think I got some serious problem with those combinatorial proofs. Why would the following be true ($1\leq r\leq k\leq n$): $$\sum_\limits{j=r}^{n+r-k}\binom{j-1}{r-1}\binom{n-j}{k-r} = \binom{n}{k}?...
2
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1answer
27 views

Finding a recurrence for number of paths in a certain tree

I have a graph which looks like this: The question is to find a recurrence for $a_n$ - the number of paths of length $n$ that start in vertex $A$. How do you tackle these kind of problems? There is ...
0
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1answer
51 views

Size of family $\mathcal F = \{F_1, \ldots, F_m\}$ is at least $\lceil \log_2n\rceil$.

A family $\mathcal F = \{F_1, \ldots, F_m\}$ of subsets of $\{1,2,\ldots,n\}$ is said to be separating if for any two elements $1 \leq i < j \leq n$, there is some set $F \in \mathcal F$ such ...
1
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1answer
48 views

Combinatorial identity $\sum_{k=0}^{n}\frac{n-k}{k+1}\binom{n}{k}^2 = \binom{2n}{n-1}$

I have an identity $$\sum_{k=0}^{n}\frac{n-k}{k+1}\binom{n}{k}^2 = \binom{2n}{n-1}$$ for which I'm looking for a combinatorial proof. Any ideas? I was thinking about separating $2n$ on boys and ...
3
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1answer
42 views

Is there a “balanced knapsacks” problem with a known result?

You're going on a trip with some friends and want to share the load of the camping gear as evenly as possible. Each of you is equally strong, and each of your knapsacks is identical. Can the fairest ...
7
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2answers
182 views

Find number of integral solutions of a*b*c*d = 600

The number of ordered solutions comes out to be 800. I need to find the number of distinct solutions but I'm stuck at calculating the possible combinations. Any ideas on how to proceed further?
2
votes
4answers
53 views

Circular Arrangement with numbers

The number of ways of arranging 2 women and 7 men around a circular table containing nine numbered chairs such that the women are not together. I am getting answer as 7!*7c2(arranging 2 women in the ...
0
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2answers
47 views

No of integral solutions to an equation confusion?

For an equation like ${ x }_{ 1 }+{ x }_{ 2 }+{ x }_{ 3 }=60$, I am seeing that in some books they are using $( (60 + 4 - 1) C (4) )$ as solution whereas in some book they are using $( (60 + 4 - 1) C (...
0
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2answers
56 views

A soccer squad contains $3$ goalkeepers, $7$ defenders, $9$ midfielders and $4$ forwards.

A soccer squad contains $3$ goalkeepers, $7$ defenders, $9$ midfielders and $4$ forwards. So I understood the first part of the question: $(i)$ In how many ways can a team of $1$ goalkeeper, $4$ ...
0
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1answer
24 views

Combinatorial interpretation of multinomial function. [closed]

Given $n$ items if we pick $k$ we use binomial function. What is the analogy with multinomial function?
7
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4answers
149 views

Find the probability that a word with 15 letters (selected from P,T,I,N) does not contain TINT

If a word with 15 letters is formed at random using the letters P, T, I, N, find the probability that it does not contain the sequence TINT. (I just made up this problem.)
1
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1answer
46 views

Notation for probability: $C_n^r$, $P_n^r$, $A_n^r$?

I was told that $C^{n}_{k}$ refers to combinations or choose k elements from n elements, $\bar{C^{n}_{k}}$ refers to combinations with repetitions (i.e. $C^{n+k-1}_{k}$), and $P^{n}_{k}$ refers to ...
3
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1answer
125 views

How many possible functions?

Take $f:\{1,2,3,4,5,6,7\}$ to $\{0,1,2,3,4\}$ How many such functions satisfy the cardinality of the pre-image of the set $\{3\}$ is equal to $3$. I thought it would be $35$, i.e :$7\choose{3}$ ...
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0answers
71 views

Colouring a Tree

There are k different colors available. How many ways are there to color each vertex of the tree in one of the k colors such that for any pair of vertices having same color, all the vertices belonging ...
6
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1answer
120 views

Picking pairs of socks from a drawer.

There are $n$ socks in a drawer, of $m$ different colours. Initially, the probability of picking a sock of colour $c_i$ at random is $\mathbb{P}(c_i) \cdot 2r$ socks are picked at random, without ...
0
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0answers
14 views

Deduce Max flow min cut from Menger's theorem

I want to deduce the max flow min cut theorem from Menger's theorem, both on arc-connectivity in digraphs. Given a network with integer capacities c, one may replace each arc a by c(a) parallel arcs ...
2
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2answers
373 views

How many ways are there to distribute pens between two girls and one guy?

There are two girls and one guy and 121 pens. How many ways are there to distribute pens between two girls and one guy, so that the girls have the same number of pens. The pens all are identical. ...
1
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1answer
30 views

Number of distinct equivalence classes of $\mathbb Z_n$ of the “ associate ” equivalence relation

Define an equivalence relation on $\mathbb Z_n$ as : For $a,b \in \mathbb Z_n $ , $a\sim b$ iff $\exists k \in U_n=\mathbb Z_n^{\times}$ such that $a=kb$ (i.e. $a,b$ are related if they are "...
1
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1answer
42 views

Summation of factorial.

$$2(\frac{1}{3!\times7!}+\frac{1}{1\times9!})+\frac{1}{5!\times5!}=\frac{2^a}{b!}$$ find $a,b$ by some predictions I see $b=10$ but what about numerator. I think we have to $\sum {N\choose r}=2^N$ but ...
1
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1answer
26 views

Finding number of subsets of set S that have r elements in common with set T

I've been going crazy trying to solve this. The question asks For some $0 \le r \le k \le n$, how many subsets of {1...n} have r elements in common with the set {1..k}. Describe two sets S and T such ...
-1
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0answers
31 views

Problem in solving a question of combinatorics. [duplicate]

In how many ways can $20$ indistinguishable pencils be distributed among four children $A$,$B$,$C$ and $D$ such that each children gets at least one pencil? My Work : At first I distribute $4$ ...
1
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2answers
71 views

In how many ways can $20$ indistinguishable pencils be distributed among four children $A$,$B$,$C$ and $D$? [closed]

In how many ways can $20$ indistinguishable pencils be distributed among four children $A$,$B$,$C$ and $D$? What is the actual technique for solving such problems?Please help me.
1
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1answer
24 views

# of bit strings of length n (even>2), with n/2-1 zeros and n/2+1 ones, zero followed by one

case 1: What is the number of bit strings of length 4, with 1 zero and 3 ones, zero must be followed by one Answer: 3 case 2: What is the number of bit strings of length 6, with 2 zeros and 4 ones, ...
-1
votes
1answer
61 views

How many of the integers are multiples of $10$ or $9$ but not a multiple of $90$? [closed]

Good evening, dear people! Who can help me with this job? Among $410$ integers, $237$ are divisible by $10$, $137$ are multiples of $9$, $53$ are multiples of $100$, $111$ are multiples of $90$, $49$ ...
1
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7answers
99 views

9 people sit in a row. 2 dressed in Red, 7 blue and 14 yellow. What is the P that at least 2 guys in yellow will sit next to a another in yellow?

1There are $\frac{21!}{2!3!} = 120$ total positions (disregarding order within same colour). I imagine labelling the people Y (yellow) and NY (not yellow), so I imagine I have $4$ copies of the letter ...
0
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2answers
62 views

the sum of all four digit multiples of 6

The sum of all four digit multiples of $6$ is equal to: A. $8~274~489$ B. $8~247~498$ C. $8~241~996$ Can you help me with this question? I've tried $$S_n= \frac{n(a_1+a_n)}{2}$$ with $...
3
votes
1answer
37 views

Uniqueness of graph neighbourhood sizes

I was thinking about graphs the other day, and had the following questions which I suppose fall under the topic of graph reconstruction. I am not very familiar with the literature, so in case this ...
0
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1answer
41 views

How to find the General expression of $\sum_{k=0}^ {\lfloor n/3\rfloor} {n \choose 3k}$ [duplicate]

Well as the title says I'm having problems trying to derive a general expression for this sum which involves cubic roots of unity $$\sum_{k=0}^ {\lfloor \frac n 3\rfloor} {n \choose 3k}$$ Need help ...
1
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1answer
58 views

Proving a polynomial with binomial coefficients has non-real roots

I have the following family of polynomials: $$p_n(x) = \sum_{k=0}^n {n \choose k} \frac{n+2}{n+2-k} x^k$$ I conjecture that it has non-real roots for all $n \geq 2$. This holds for all the small ...
1
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1answer
85 views

Counting the number of partitions

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided ...
0
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1answer
42 views

Characterization of bicycle graphs

By "bicycle graph" I mean a minimal connected simple graph with at least two cycles. From Wikipedia: There are three possible types of bicycle: a theta graph has two vertices that are connected ...
3
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0answers
91 views
+50

Maximal unit lengths in 3D with $n$ points.

Given $n$ points in 3D space (V), what is the maximal number of unit distance lengths (E) between those points? Here are a few possibilities. Some of them are chromatic spindles. ...
1
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1answer
37 views

Finding the combination where p of the items are identical.

Suppose we have $n$ objects in which $p$ items are identical. Of course, $n-p$ elements are distinct. Then what is the combination of $n$ objects taken $r$ at a time? That is, what is $C(n,r)$, but ...
1
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0answers
58 views

What is the Star of David theorem?

I came across a MathWorld entry for the Star of David Theorem, but it doesn't provide much context. I have never heard of this before, can somebody explain its significance and any applications it ...
7
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3answers
143 views

How many integer solutions are there of the equation $|x_{1}|+|x_{2}|+\cdots +|x_{k}|=n$?

How many solutions are there to the equation $$|x_{1}|+|x_{2}|+\cdots +|x_{k}|=n$$ for $n,k\in \mathbb N$ and $\forall\ 1\leq i\leq k,\ x_{i}\in \mathbb Z$? Any ideas? I don't know how to ...
5
votes
1answer
80 views

Upper bound on the minimum distance between $N$ points chosen inside the unit circle?

I guess this is a well-known problem but I'm not sure where to find it on the web. $N \ge 2$ points are chosen in the interior or the boundary of the unit circle. What is the best upper bound on the ...
0
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0answers
28 views

Solutions to Diophantine Moving Window Inequations

I am interested in finding the number of non-negative integer solutions, $N(m,h,u)$, to this system of inequations $$ \left\{ \matrix{ 0 \le x_{\,0} + x_{\,1} + \cdots + x_{\,m} \le u \hfill \...