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
13 views

Representing geometric series as sum of binomial coefficients

I was learning generating functions, where faced following: $$(1-x)^{-n}=\sum_{i=0}^\infty\binom{n+i-1}{i}x^i$$ I didn't get how is this arrived. I tried out to prove this to me. But failed. ...
0
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0answers
17 views

How many ways to pile up boxes in a direction

Lets assume that there are columns with spesific limits, and there are boxes on these columns. We need to find all the possible ways(positions) from the original layout to the layout that completely ...
0
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1answer
19 views

Let $n = p_1^{k_1} + p_2^{k_2} + … + p_m^{k_m}$ how many ways can $n$ be written as a product of two positive integers

Let $n = p_1^{k_1} + p_2^{k_1} + ... + p_m^{k_m}$ where $p_1, p_2,...,p_m$ are distinct prime numbers and $k_1,k_2,...,k_m$ are positive integers. How many ways can $n$ be written as a product of two ...
4
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1answer
56 views

Why does the Möbius function take its values so often in $\{0,+1,-1\}$?

The Möbius function of a locally finite poset $P$ is defined on its intervals $[x,y] \subseteq P$ recursively by $$\mu([x,x])=1$$ $$\forall x < y : \mu([x,y])=-\sum_{x \leq z < y} \mu([x,z])$$ ...
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0answers
11 views

Finding the maximal product of numbers of permutations

Let $n\geq 1$ be a total number of objects that must be taken from $m\geq 1$ sets of objects. For all $i \in \{1,\cdots,m\}, \ M_i \in \mathbb{N}^*$ denotes the number of objects present in the set ...
3
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1answer
33 views

How many ways we can choose items from different boxes

I searched through the internet but couldn't find my answer, which can either be a very simple or a hard one. Assume there are $3$ boxes, which carry, respectively, $1$, $4$, $2$ items. My question ...
-2
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0answers
33 views

Expected Value of a Mangoes

There is a $N \times N$ grid. Each square in the grid either has or does not have a mango tree. For example, suppose the field looks as follows: ...
1
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1answer
31 views

Selection of subsets

This is an supplementary exercise from Miklos Bona: A walk through combinatorics. We want to select as many subsets of $[n]=\{1,2,3,..,n\}$ without selecting two subsets such that neither one of them ...
1
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4answers
50 views

To prove $\sum_{n=0}^\infty \binom{r}{x}\binom{N-r}{n-x}=\binom{N}{n}.$ [duplicate]

To prove $$\sum_{x=0}^n \binom{r}{x}\cdot \binom{N-r}{n-x}=\binom{N}{n}.$$ I tried comparing the coefficients of $(1+x)^{(n+k)} = (1+x)^n(1+x)^k$ but couldn't reach the answer.
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3answers
47 views

Number of subsets without objects being adjacent

I have a set of $20$ people who have names. 1) How can I count the different subsets of size $3$, where the people of that subset are not allowed to be next to each other in an (alphabetically) ...
1
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1answer
41 views

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ [duplicate]

Find a closed formula for $\sum_{n=1}^\infty nx^{n-1}$ I am trying to use the derivative of generalized binomial theorem, $\frac{d}{dx}[(x+1)^r=\sum_{n=0}^\infty \binom{r}{n}x^n] ...
2
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0answers
26 views

Decorating a single flower box with $4$ spaces with $4$ differently colored flowers, so that there is no flower box with more than $1$ yellow flower.

What we are actually talking about Available flowers: Pink, Red, Yellow and Blue ($4$ distinct kinds). Available spots in the flower box: $4$. Restriction: Max. $1$ yellow flower in the flower box. ...
7
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3answers
59 views

Prove $\binom{3n}{n,n,n}=\frac{(3n)!}{n!n!n!}$ is always divisible by $6$ when $n$ is an integer.

Prove $$\binom{3n}{n,n,n}=\frac{(3n)!}{n!n!n!}$$ is always divisible by $6$ when $n$ is an integer. I have done a similar proof that $\binom{2n}{n}$ is divisible by $2$ by showing that ...
0
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0answers
47 views

Number of integral solutions to $y_1 + y_2 + y_3 + y_4 =30$ with $y_i \ge 2$

How many solutions there are to the given equation that satisfy the given condition: $y_1 + y_2 + y_3 + y_4 =30$, each $y_i$ is an integer that is at least $2$. I don't know how to start this ...
0
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1answer
25 views

Suppose that after a five-card hand is drawn, the cards in it are put back in the deck and another five-card hand is drawn.

Suppose that after a five-card hand is drawn, the cards in it are put back in the deck and another five-card hand is drawn. a) What is the probability that the two hands have no card in common? b) ...
0
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1answer
18 views

Recurrence for the number of n tuples with restrictions

If $a_{n}$ is the number of $n$ tuples $(b_{1}, b_{2},...b_{n})$ with $b_{i} \in[4]$ that have at least one 1 and have no 2 appearing before the first 1. What is the recurrence for $a_{n}$?
1
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1answer
33 views

Finding the no of ways to count the letters in an English alphabet

How many strings of six lower case letters from the English alphabet contain a) the letter $a$? b) the letters $a$ and $b$? c) the letters $a$ and $b$ in consecutive positions with $a$ preceding ...
0
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0answers
25 views

Restricted integer composition, where every summand belongs to a different set

Could I get any suggestion on how to calculate the number of restricted integer compositions of a number $n$ with $k$ parts, where every summand belongs has its own subset? \begin{equation} ...
0
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1answer
41 views

Is there an equation to find out how after $\frac{6!}{6}$ to locate clockwise increase in numbers in sets of 2

So I asked this question last night what is the max possible combinations of 1 2 3 4 5 6 without repeating And as stated I don't know what symbols mean, but I learned what $!$ is and how it works ...
0
votes
1answer
26 views

Number of Unique Ranks of High Card in Three Card Brag

Well the game is called Teen Patti in India. Almost similar to Three Card Brag a British game. There are total $16440$ Unique High Card hands are present. (Considering the suit.) Hand $1 = 5$ Heart, ...
1
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1answer
13 views

Express reverse inversion, major index, descents in terms of the forward direction.

Given $w=a_1a_2...a_n \in S_n $, then the reverse of $w$ is $w^r=a_n....a_2a_1$. Express inv($w^r$), des($w^r$) and maj($w^r$) in terms inv($w$), des($w$), maj($w$), respectively. I know the ...
2
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0answers
44 views

upper bounding alternating binomial sums

So we know that $\sum_{i=0}^t\binom{m}{i}\binom{n-m}{t-i}=\binom{n}{t}$ by a simple counting argument. Now is there any bound on the quantity $\sum_{i=0}^t(-1)^i\binom{m}{i}\binom{n-m}{t-i}$? Can we ...
0
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1answer
32 views

A box with $3$ types of colored balls.

In a box there are $15$ white balls, $8$ black balls, and $12$ red balls. We extract $6$ balls, without putting them back. $(a)$ What is the probability that the first ball is red, the second and ...
0
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1answer
17 views

Morse code symbols represented by sequences of seven or fewer dots and dashes

In Morse code, symbols are represented by variable length sequences of dots and dashes. (For example, A = · −, 1 = · − − − −, and ? = · · − − · ·.) How many different symbols can be represented by ...
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3answers
44 views

An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon.

An ice-cream shop sells $11$ kinds of ice-cream, including mango and lemon. For a bowl, one chooses at random five kinds (not necessarily different). $(a)$ How many different bowls can be made? ...
0
votes
1answer
14 views

A finite, undirected, connected and simple graph with Eulerian circuit has $3$ vertices with the same degree

Let $G=(V,E)$ a finite, undirected, connected and simple graph, $|V| \ge 3. \space$ Prove: If $G$ has Eulerian circuit then $G$ has $3$ vertices with the same degree.
2
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3answers
41 views

If $n_{1}+n_{2}+n_{3}+n_{4}+n_{5} = 20.$ Then number of such distinct arrangements of $(n_{1},n_{2},n_{3},n_{4},n_{5})$

Let $n_{1}<n_{2}<n_{3}<n_{4}<n_{5}$ be the positive integers such that $n_{1}+n_{2}+n_{3}+n_{4}+n_{5} = 20$ Then number of such distinct arrangements of ...
3
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4answers
50 views

How many $3$-tuples $(a, b, c) \in M^3$ are there such $a+b+c$ is even?

The task is the following: $M= \left \{ 1,2, ... 99,100 \right \}$ How many $3$-tuples $(a, b, c) \in M^3$ are there such $a+b+c$ is even? I tried to solve it this way: There are only two ...
1
vote
1answer
80 views

What is so special about Higman's Lemma?

Is there a motivational example of Higman's Lemma that brings out the true beauty and importance of Higman's Lemma? What is the thing that made so many people care about it? For an example, I was ...
0
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1answer
48 views

When is the order important in Combinatorics?

In a shop five different type of chocolates are sold. How many different ways 6 chocolate bars can be chosen in such a way that at least 3 chocolate bars must be of type one and at most one of type ...
0
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0answers
35 views

Here is a question on combinatorics [on hold]

here are ten items on sale at a bazaar, each costing less than one dollar. Prove that it is possible for two people to purchase distinct subsets of these objects and pay exactly the same amount. (Not ...
0
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0answers
11 views

Bounding entries of random vector

Given a random vector $\mathbf{e} \in \mathbb{R}^n$, is it possible to count (or bound) the number of entries in $\mathbf{e}$ that have $|e_i| \ge 1/ \sqrt{n}$? It is known that entries in ...
1
vote
1answer
109 views

Covering board with pieces

Suppose we have board, of size (16x16) And 31 (1x4) + 33 (2x2) pieces. Is it possible to cover up board with those pieces, if so - how? If not - why? So far I was unable to think of anything ...
6
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0answers
27 views

Chromatic Number of Circulant Graph

Consider the Circulant Graph $Ci_{2n}(1,n-1,n)$ as described here: http://mathworld.wolfram.com/MusicalGraph.html Another way to describe $Ci_{2n}(1,n-1,n)$ would be $2n$ vertices with vertex set ...
1
vote
2answers
37 views

$6$ real numbers, sum of any $3$ consecutive is negative, while sum of any $4$ consecutive is positive. Prove false. [on hold]

It's from my combinatorics class, could anyone give me some hints? Thanks Sorry, I shortened the original phrasing of the question, which made it ambiguous here. The question goes: A computer ...
0
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0answers
10 views

Families of 3-element subsets such that no two intersect more than once

Another user asked the following question: "How can I determine the size of the largest collection of $k$-element subsets of an $n$-element set such that each pair of subsets has at most $m$ elements ...
0
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2answers
26 views

Permutations with repeated item [on hold]

I have: Orange Apple Orange Guava Pineapple Watermelon Strawberry Note the repeated Orange. Out of these 7, I have to choose 4 fruits. How many permutations are possible?
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0answers
24 views

equal unions and intersections

Let $N$ be a $n$-element set and $k\ge n+2$. The sets $P_1,\dots,P_k$ are nonempty and their union equals $N$. Then there exists disjoint sets $I,J\subset\{1,\dots,k\}$ such that $\bigcup_{i\in ...
2
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1answer
72 views

Coupon Collectors Problem with Packets: Clarifying Wikipedia

The Coupon Collector's Problem (CCP) is very useful in many applications. However, the "default" CCP is relatively simple: suppose you have an urn containing $n$ pairwise different balls. Now you want ...
3
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4answers
82 views

Given that $6$ men and $6$ women are divided into pairs, what is the probability that none of the women will sit with a man?

I've generalized the question I was given here for simplicity: $6$ men and $6$ women are to be paired for a bus trip. If the pairings are done randomly, what's the probability that no women will end ...
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0answers
17 views

Confusing Conditional Probability question 68 [on hold]

The four top tennis players in the world A, B, C, and D are invited to a special tournament where the winner gets one million dollars. In round one, Player A plays player D and player B plays player ...
2
votes
4answers
90 views

Prove the formula $\sum_{k=1}^n k\binom{n}{k} = n \cdot 2^{n-1}$ for all integers $n > 0$ [duplicate]

I just got to this question and I became a question mark. I wonder if anyone can help me with this one, because I don't even know how to begin to tackle this problem. The question: Prove the ...
2
votes
1answer
51 views

In how many ways can $8$ appointments be scheduled for a physician visiting a nursing home with $20$ patients? [on hold]

A physician routinely visits a local nursing home on Thursday mornings to examine patients. Suppose the facility has $20$ residents, but the physician only has time to check $8$. The supervisor places ...
0
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0answers
4 views

Counting subgraphs of bounded extremal degrees

Let $m\leq n-1$. Is there a closed expression counting the subgraphs of minimum degree $\geq m$ (resp. maximum degree $\geq m$) on $n$ labelled vertices?
0
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1answer
40 views

Finding a closed formula for: $1\cdot2\cdot3+2\cdot3\cdot4+…+(n-2)\cdot(n-1)\cdot(n)$ [duplicate]

As I calculated the sum of the serie above doesn't exist(sum doesn't converge). How can I prove it using the double computing(combinatorical method)?
1
vote
1answer
31 views

How many ways to divide $n$ different pieces of chocolate in two non empty groups?

After the example I think that the order of the groups doesn't matter so ${(A),(B,C)}$ and $(B,C),(A)$ counted as $1$. Suppose we split $5$ chocolates into a group of size $1$ and a group of size of ...
-1
votes
5answers
81 views

Deck of Cards Stats Probability Question [on hold]

Randomly select two cards in sequence from a full deck of 52 cards, what i s the probability that the first one is a King given that the second one is a King. If someone can please help me with this ...
2
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0answers
27 views

Distribution of distinct object problem

So i was given this question. How many ways are there to place 10 distinct people within 3 distinct rooms with exactly 5 people in the first room and 2 people in the second room? So i asked this ...
0
votes
1answer
17 views

Summation of all j-combinations (Expanding composition formula)

I found a formula for a problem that I was trying to solve, the Formula 3.2 in Section 3 at page 441 of this document.I am a little unsure about the "Summation over all j-combinations". Here is what I ...
2
votes
0answers
32 views

Generating subsets with 1 common element

I have a number $n$ and a set $S$ of $n(n-1)/2$ elements : $ \{1, 2, \ldots, n(n-1)/2\}$ I'm looking for an algorithm to generate $n$ distinct subsets of $S$, each having $n-1$ elements, with the ...