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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1
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1answer
21 views

Computing $C_0^2+C_1^2+C_2^2+C_3^2+ \cdots +C_n^2$

If $C_k$ denotes binomial coefficient of choosing $k$ objects from a set of $n$ objects how to calculate this: $$C_0^2+C_1^2+C_2^2+C_3^2+\cdots +C_n^2$$
-1
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0answers
21 views

Combinatorics Review; Discrete Math

I have forgotten a lot of the counting portion of my discrete structures course and need some explanations how to count, maybe some general strategies on counting. Some example questions I need ...
0
votes
1answer
19 views

Problem Solving Involving Permutation

Find the number of 6-digits number with no 3 consecutive number with same digits. Note that 0 might be the first number. I have tried to find the number with no pairs, 1 pairs, 2 pairs and 3 pairs. ...
2
votes
1answer
17 views

Counting Review; Discrete Structures

I have forgotten a lot of the counting portion of my discrete structures course and need some explanations how to count, maybe some general strategies on counting. Some example questions I need ...
0
votes
1answer
16 views

Counting Problem; Discrete Structures

I have forgotten a lot of the counting portion of my discrete structures course and need some explanations how to count, maybe some general strategies on counting. Some example questions I need ...
0
votes
0answers
14 views

How should i go about proving an expression of this kind?

Lets say i have a complete bell polynomial that is equal to a summation such that $$ B_n(d_1,d_2,\cdots,d_n) = \sum_{k=0}^{n}[g(x)^{-k} h(k)] $$ Where $d_n = \frac{d^n}{dx^n}[f(x)\ln(g(x)]$ and ...
0
votes
1answer
30 views

Counting Question; Discrete Structure

I have forgotten a lot of the counting portion of my discrete structures course and need some explanations how to count, maybe some general strategies on counting. Some example questions I need ...
1
vote
1answer
20 views

Please explain counting; Discrete Structures

I have forgotten a lot of the counting portion of my discrete structures course and need some explanations how to count, maybe some general strategies on counting. Some example questions I need ...
1
vote
1answer
28 views

Combinatorics- monotonic subsequence

For each natural number $n$, find a sequence of $n^2$ real numbers which contains no monotonic subsequence of more than $n$ terms. I've been stuck on this for a while. Can somebody please point me ...
-3
votes
0answers
11 views

Problem in Kernerl of Digraphs [on hold]

Prove that every digraph that is not kernel perfect contains a critical kernel perfect digraph. Where: A kernel K of a digraph D is a subset of D that satisfies: (1) Every pair of vertex in this ...
1
vote
1answer
17 views

Find the chance that subset $B$ is distributed evenly between $(A_1,A_2,A_3)$

We are given set $A$ which is divided to the 3 parts ($A_1$,$A_2$, $A_3$). $|A| = n = 9k$. For $i,j = 1,2,3;\space \forall i \ne j : A_i \cap A_j = \emptyset; \space\space |A_i| = \frac n 3$. ...
2
votes
1answer
26 views

Combinatorics, dividing objects into groups.

Assuming we have got 5 horses, that are competing in a race, and assuming 2 different horses can arrive at the exact same time. How many possibilities there are for outcomes? for 3 horses for example ...
3
votes
1answer
69 views

Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
0
votes
0answers
28 views

Multi-ruled combinatorics problem (need this for my lab)

I need to know this for practical purposes and not homework, learning etc.. Say I have 3 electrodes A,B and C. Say I also have 3 electrolytes A,B and C. If electrode A has to be in electrolyte A, ...
-2
votes
2answers
54 views

Combinatorics homework problem [on hold]

In how many ways can $23$ different books be given to $5$ students so that $2$ of the students will have $4$ books each and the other $3$ will have $5$ books each?
4
votes
2answers
58 views

How many words can be written with $aabbbccdd$ such that no two equal letters are adjacent?

I'm trying to count this using the principle if inclusion-exclusion. I've done the following: Counting the number of permutations of $aabbbccdd$. $9!$ Counting the number of ...
0
votes
1answer
29 views

how to place k rooks on the shaded squares of a m×n grid-like board

I’m given a m×n grid-like board and there are some shaded squares in every row of the board. I have to place one or more rooks on the shaded squares in such a way that no two rooks attack each other. ...
0
votes
1answer
30 views

How many ways can one “fit” $m$ non-overlapping sub-segments of length $k$ into a segment of length $n$?

Given discrete one-dimensional space (a "segment") of length $n$, how many ways can one fit a $m$ non-overlapping sub-segments of length $k$ in this space? This seems like a very simple question, but ...
1
vote
0answers
12 views

Derangement of multiset using recursive relation

Recently,I have read articles on derangement but now I want know about how to derange a multiset. By using inclusion-exclusion one can find out the number of ways to derange a multiset. I'm looking ...
1
vote
2answers
37 views

let $D_n$ be the number of permutations of $\{1,2,3,…n\}$ which leave no element fixed.

Let $n\geq2$ and let $D_n$ be the number of permutations of $\{1,2,3,\dots,n\}$ which leave no element fixed. How to write an expression for $D_n$ in terms of $D_k$? I don't know how to start. Please ...
0
votes
0answers
20 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 ...
0
votes
0answers
10 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 ...
1
vote
1answer
96 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 ...
0
votes
0answers
20 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))] = ...
1
vote
1answer
37 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 ...
0
votes
1answer
24 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 ...
0
votes
2answers
20 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?
0
votes
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?
0
votes
1answer
46 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 :( ...
0
votes
0answers
47 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 ...
1
vote
1answer
44 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} ...
13
votes
6answers
533 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} ...
6
votes
0answers
61 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 ...
0
votes
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
votes
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 ...
0
votes
0answers
19 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 ...
0
votes
0answers
39 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
votes
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
235 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
votes
1answer
48 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
votes
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
0
votes
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
votes
3answers
55 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 ...
1
vote
1answer
27 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
votes
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
vote
3answers
70 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 ...
1
vote
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
69 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
votes
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 ...