This tag is for basic 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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13 views

Question about some properties of combinatorial structures

Consider $\mathcal A$ as the set of perfect matchings in the complete bipartite graph $K_{n,n}$ and let $i$ be an edge of $K_{n,n}$. Let $$ B_i=\{a\in \mathcal A: \hbox{matching }a\hbox{ has edge ...
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0answers
7 views

Idempotent generators of the four binary QR codes of length 7

I have a coding theory assignment and I thought it would be a good idea to double check before I hand it in. I'm asked to find the idempotent generators of the four binary QR codes C1, C2, C3, C4, of ...
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1answer
35 views

A sequence of $n^2$ real numbers which contains no monotonic subsequence of more than $n$ terms

I'm following a Combinatorics course at the moment, and have recent proved the Erdős–Szekeres Theorem (or, at least, some variation of): A sequence of length $n^2 + 1$ either contains an ...
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0answers
18 views

Do these statements prove this formula?

$$ \frac{d^n}{dx^n}[g(x)^{f(x)}] = g(x)^{f(x)} B_n(d_1,\cdots,d_n) $$ Calling $$ d_n = \frac{d^n}{dx^n}[ln(g(x))f(x)] $$ Since faa di bruno's formula states $$ \frac{d^n}{dx^n}[f(g(x))] = ...
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1answer
20 views

Unique permutations from set with repetitions

I am new to combinatorics and might ask a trivial question: There are $69$ different items, each present $4$ times. From this total of $276$ items, $20$ should be picked at random. I need the formula ...
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1answer
17 views

Is my idea of incoming/outgoing arcs correct?

I'm reading Jungnickel's Graphs, Networks and Algorithms. I've met the following lemma: I know that $e^{-}$ are the incoming vertices and $e^{+}$ are the outgoing vertices. Then I've tried to ...
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2answers
19 views

There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist-

There are eight males and 12 females in a certain club. In how many ways can a committee of five be chosen if it is to consist Entirely of Males? Entirely of Females? 2 males and 3 females?
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1answer
26 views

In how many distinct ways can a group of letters be ordered? [on hold]

In how many distinct ways can the letters aaabbbbb and aaabbbbbcccc be ordered?
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1answer
28 views

Combinatorics-graph colouring [duplicate]

Show that if $K_9 $is coloured red and blue and contains no red triangle and no blue $K_4$, then every vertex must have red degree $3$ and blue degree $5$. I have absolutely no idea how to proceed :( ...
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0answers
45 views

Combining 2 numbers into a uniqe number

I am stumped on a problem, I have a set of numbers (lets say 2 numbers) A and B and i want to combine them into a unique number C where C is not reproducible by any other set thats not identical ...
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1answer
41 views

How prove this number of the methods is this $\prod\prod 4\cos^2{\frac{j\pi}{m+1}}+4\cos^2{\frac{k\pi}{n+1}}$

Question: show that an $m$-by-$n$ chessboard can be partitioned some $1$-by-$2$ the numbers of methods is $$\prod_{j=1}^{\lfloor\dfrac{m}{2}\rfloor}\prod_{k=1}^{\lfloor\dfrac{n}{2}\rfloor} ...
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5answers
420 views

Combinatorial identity with sum of binomial coefficients

How to attack this kinds of problem? I am hoping that there will some kind of shortcuts to calculate this. $$\sum_{k=0}^{38\,204\,629\,939\,869} ...
5
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0answers
45 views

Number of sets of vertices whose union of neighbours contains exactly $k$ vertices

Suppose a bipartite graph $g$ consisting of $2n(n-1),n\in\Bbb N,n>1$ vertices, is divided equally into two colors: red and blue, and is constructed as follows: For example, $g$ for $n=3$: If ...
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2answers
28 views

Possible 4 character passwords involving a letter and a digit.

A password consists of 4 characters, each of which is either a digit or a letter of the alphabet. Each password must contain at least ONE digit and AT LEAST ONE letter. How many different such ...
0
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2answers
24 views

Probability of an event happening while another doesn't

Say you have a bag with $5$ numbers $(1,2,3,4,5)$. What is the probability that I will draw a $1$ if I draw $3$ times (no replacement)? What is the probability that I will draw a $1$ if I draw 3 ...
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0answers
17 views

dimension of vector space $\frac{\langle e_{ab_1\ldots b_p}\rangle}{\langle \sum_{1\leq i\leq p}e_{ab_1\ldots \widehat{b_i}\ldots b_pc}\rangle}$

Let $p$ be a prime and $n\!\in\!\mathbb{N}$. What is the dimension of the $\mathbb{Z}_p$-module $$V_{p,n}=\frac{\langle e_{ab_1\ldots b_p};\: 1\leq a<b_1<\ldots<b_p\leq n\rangle}{\langle ...
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0answers
38 views

Presentation of 2 images in a random but counterbalanced way

Problem: For 18 trials randomly a ‘left’ labeled image or ‘right’ labeled image is shown. The first 9 trials should contain the opposite number of left images as the last 9 (a.k.a. counterbalance). ...
0
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1answer
26 views

How many different teams can be created between two groups?

If a company has 8 painters and 12 electricians. How many different teams can be created with 1 painter and 1 electrician? I know that the number of ways a team can be made is: $ {8 \choose 1} * ...
2
votes
3answers
229 views

Probability of no ace in a 6 card hand, given 4 are not aces.

A player is dealt six cards out of a normal deck of cards. He looks at the first four and notices there is no ace among them. What is the probability that he does not have an ace at all. This sounds ...
2
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1answer
45 views

Find the chance that $a^3 + b^3 \equiv 0 (\mod 3)$

We are given set of integer numbers $\{1,2, \dots N\}$. $N \ge 3$ Then perform a drawing with replacement of two elements $a$ and $b$. Problem is to find the probability of following statement holding ...
-2
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1answer
18 views

number of possible outcomes in a license plate with conditions [on hold]

howmany license plates can me made when a) first two letters are different and the rest different digits e.g. DA3457 b) two letters in alphabetical order and the digits increasing e.g. CD1234
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2answers
21 views

Story proof for choosing a committee

Give a story proof that $\sum_{k=0}^n k{n\choose k}^2 = {n{2n-1\choose n-1}}$ Consider choosing a committee of size n from two groups of size n each , where only one of the two groups has people ...
4
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3answers
53 views

possible pizza orders

You are ordering two pizzas. A pizza can be small, medium, large, or extra large, with any combination of 8 possible toppings (getting no toppings is allowed, as is gettting all 8). How many ...
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1answer
25 views

Choosing schedule for courses

To fulfill the requirements for a certain degree, a student can choose to take any 8 out of a list of 20 courses, with the constraint that at least 1 of the 8 courses must be a statistics ...
2
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1answer
40 views

A problem on distributing indistinguishable balls into 10 different groups such that…

I got this problem which I am stuck at for an hour and half: Suppose that we have an infinite number of indistinguishable balls and we need to distribute them into 10 different groups such that $ ...
0
votes
2answers
13 views

How many different pairs can I have from two groups?

A company has 8 painters and 12 electricians, and teams can be created of one painter and one electrician. How many different teams can be created? My best guess is: $ {8 \choose 1} * {12 ...
1
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3answers
68 views

Olympic elementary combinatorics problem

This is a problem taken from the regional selections of the Italian mathematical olympiads: A knight is placed on the bottom left corner of a $ 3\times3 $ chess board. In how many ways can you move ...
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4answers
34 views

Select one or zero elements from a set

I am far from a mathematician. Still. I want to formally express that only 0 or 1 element of a series of sets (1...n) is selectet to form a new set. Example: I have three sets $S_1 = \{1,2,3\}$, $S_2 ...
3
votes
0answers
66 views

Why does $n$ always divide this sum?

If we assume $m=p_1^{a_1}\cdots p_s^{a_s}, n=p_1^{b_1}\cdots p_s^{b_s}p_{s+1}^{b_{s+1}}\cdots p_t^{b_t}$, where $0<a_i<b_j$, $p_j$ are different primes($i=1,\cdots,s; j=1,\cdots, t$). Then ...
3
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0answers
40 views

In how many ways 3 persons can solve N problems.

There are $3$ friends $(A,B,C)$ preparing for math exam. There are $N$ problems to solve in $N$ minutes. It is given that: Each problem will take $1$ minute to solve. So all $N$ problems will be ...
2
votes
3answers
35 views

question on morse code

The morse code is made up of marks called dots and dashes."Q", for example is (--,--).Is it possible to make up such a code so that every letter of the alphabet is represented by at most three marks? ...
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0answers
38 views

Proving that elementary row operations are preserved after multiplication

If $E$ is an elementary $n \times n$-matrix, show that if $A$ is any $n\times n$-matrix, then $EA$ is a matrix obtained by carrying out a single elementary row operation on $A$, and that $AE$ is a ...
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0answers
14 views

In a best-of-7 match possibilities for 7th game win

In a best-of-7 match A vs B, where the match will end as soon as either player has 4 points. How many possible outcomes for the individual games are there, such that the match lasts for 7 games and A ...
1
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2answers
26 views

pairing possibilities in chess game

There are 20 people at a chess club on a certain day. They each find opponents and start playing. How many possibilities are there for how they are matched up, assuming that in each game it does ...
1
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1answer
24 views

Number of arrangements of the word “MAMMAL” where M is not together

This is in reference to this question. Letter Arrangement with Permutations _A_A_L_ IF M is not together, then M can go into 4 distinct places (denoted by the underscores above). So the number of ...
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1answer
42 views

Probability that a monkey at a type writer types “hamlet” [duplicate]

A monkey types each of the 26 letters of the alphabet exactly one time. What is the probability that the world "hamlet" appears somewhere in the string of letters?
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1answer
16 views

Does the maximum cut implies the minimum flow?

Is it possible to reverse the result of the min-cut max-flow theorem and obtain the result that if you have the maximum cut, then you have the minimum flow? I've been thinking about it, but I have no ...
1
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1answer
27 views

What is the probability of not rolling any given number on 10 rolls of a die?

In other words, ALL combinations which don't contain at least one of the number from 1-6 would count. So for example... 5, 2, 3, 3, 4, 1, 5, 5, 3, 1 would be counted because there is no 6 Also 5, ...
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1answer
64 views

Better Explanation for an already posted question [duplicate]

Can anyone explain why in this question the answer is 5! * 2! * 10P3? I understand the 5! and 2! but for 10P3 the first thing I thought of was 3! Thanks.
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0answers
10 views

Upper bound for graphs with no k-cliques

We know that for random graphs $G(n,p)$ we have: $P[X=0]\leq e^{-\Theta(E[X])}$ where $X$ denotes the number of k-cliques in the random graph. Can this fact be used to say anything about the number of ...
-2
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1answer
33 views

Combinatorial Argument Proof

Prove: $c(40,5) = c(17,5) + c(17,4) + c(23,1) +...+ c(23,5)$ where c is the binomial coefficient. Can I use a combinatorial argument to prove?
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1answer
13 views

Distributing dimes to 2 groups of people such that each member of one group gets at least one

I have a study question that I have the answer for, but I just can't understand how or why it is the answer. The question is: $n$ dimes are distributed to $y$ young people and $o$ old people. Every ...
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0answers
21 views

Point of most theoretical potential moves in a game of Scrabble

I was recently playing a game of scrabble with a friend and the point difference all but ensured that I was going to lose (100+ points with one rack of tiles left, and no more in the "pot" and I ...
1
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1answer
17 views

Power set ordered by sum and Dijkstra shortest path

I've needed to enumerate the power set ordered by the sum of elements in each subset. Luckily I've found a nice solution here: Algorithm wanted: Enumerate all subsets of a set in order of increasing ...
0
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1answer
16 views

Number of graphs with M edges that does not contain K-clique

If we consider the space of graphs $G(n,M)$ with $n$ vertices and $M$ denotes the number of edges. Is there any way of upper bounding the number of graphs in this space that does not contain any ...
1
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1answer
19 views

Number of integer coefficient multilinear polynomials

I am looking for an expression for number of multilinear polynomials of degree atmost $t$ in $n$ variables with integer coefficients having coefficient size atmost $|B|$. Is ...
1
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1answer
18 views

possible outcomes for round-robin tennis tournament

A round-robin tournament is being held with n tennis players; this means that every player will play against every other player exactly once. How many possible outcomes are there for the tournament? ...
0
votes
1answer
14 views

Counting - possible schedules for dinner

Fred is planning to go out to dinner each night of a certain week, Monday through Friday, with each dinner being at one of his ten favorite restaurants. How many possibilities are there for Fred's ...
0
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2answers
21 views

Find a probability of $L_\sigma(A) = F_\sigma(B)$

We are given set $\{1, 2, \dots n\}$ and some random permutation $\sigma$ of that set. Sets $A, B \subseteq \{1, 2, \dots n\}$ and |$A \cap B| = 1$ and $|A| = |B| = k$ We define $L_\sigma(A)$ as the ...
1
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1answer
42 views

Maximal number of colours in a palette that allows for fewer than 500 mixtures

An artist is planning on mixing together any number of different colours from her palette. A mixture results as long as the artist combines at least two colours. If the number of possible mixtures is ...