-2
votes
0answers
34 views

mathematics solve [duplicate]

Please can you help me with this question? Out of 120 customers that visited a supermarket in a day. 62 bought clothing materials, 51 bought provisions and 48 bought kitchen utensils. 15 customers ...
0
votes
1answer
63 views

Combination Problem with Sofa [on hold]

Suppose we have 5 sofa on room A. in this room, 4 students seated on these sofa. These Strudents go to another room for eating dinner, and after that come back to room A. how many way the students can ...
0
votes
1answer
27 views

Combination with repition, Representation techniques.

Consider the following Question:A bagel shop has onion bagels, poppy seed bagels, egg bagels, salty bagels, pumpernick bagels, sesame seed bagels, raisin bagels, and plain bagels. How many ways are ...
1
vote
1answer
14 views

Eccentricity of vertices in a graph when eccentricity of one vertex is given

I have a very basic doubt. If a vertex in any graph has the eccentricity two, then what can be concluded about eccentricities of other vertices in graph. Is the eccentricity of every vertex is less ...
1
vote
1answer
28 views

How many closed binary operations on A have x as the identity?

There is this one example in my book that explains how to do this, but it's very obsecure and I just can't follow it. It says if: A = {x,a,b,c,d} then there are 5^16 closed binary operations on A ...
1
vote
3answers
60 views

How many ways to arrange $6$ children in $4$ bedrooms if at most $2$ kids per room

If I have $6$ children and $4$ bedrooms, how many ways can I arrange the children if I want a maximum of $2$ kids per room? The problem is that there are two empty slots, and these empty slots are ...
1
vote
0answers
23 views

Weak compositions with bounded partial sums

Is there an easy way to count the number of weak m-compositions of n whose partial sums are lower bounded by some function? Example: Let K be a weak 3-composition of 4 K = (k1, k2, k3) Let s(t) be ...
0
votes
2answers
57 views

How many positive integer solutions are there to the inequality $x_1+x_2+…+x_r\le n$?

The original problem is there are $r$ identical boxes and $n$ identical balls. Every box is nonempty. Then how many ways of putting balls in boxes? It is equivalent to the problem of finding ...
0
votes
0answers
20 views

Probability of x pocket pairs at a table of n people (NLHE)?

With n people at a table, what is the probability that x of them are dealt pocket pairs? There are several easy ways to approximate this but I was wondering there was an elegant solution. Any takers?
9
votes
1answer
92 views

Blocking lines of length $5$ in a $7 \times 8$ matrix; how can we know the solutions have a specific form?

A friend shared with me the following puzzle he encountered in a Chinese math competition: In a $7 \times 8$ matrix, we place tokens so that any straight line of length $5$ (horizontal, vertical, ...
0
votes
1answer
27 views

Small remarkable matroids

I'm working on a problem involving matroids $M=(E,\mathfrak{C})$ (here $E$ is the ground set, $\mathfrak{C}$ the set of circuits) with a "small" ground set $E,$ in the sense that $\sharp(E)\leq7$ I ...
3
votes
0answers
28 views

Does there exist Latin square critical sets for which deleting any entry results in arbitrarily many completions?

For those familiar with Latin squares terminology, I'll get straight to the point: Q: For all $N \geq 2$, does there exists a critical set $C$ (for a Latin square of any finite order) such that ...
0
votes
0answers
29 views

Inversion and permutations

Let call two arrays A and B with length n almost equal if for every i (1 <= i <= n) CA(A[i]) = CB(B[i]). CX[x] equal to number of index j (1 <=j <= n) such that X[j] < x. For two ...
1
vote
1answer
72 views

Expected value over many trials

I am a poker player and was talking to my friend about expected value. He claimed that if you play far enough above your bankroll, expected value can be negative, even if you have a skill edge. I ...
0
votes
0answers
39 views

The sum of palindromes from 100 to 900

I'm working with palindromes from $100-999$. I'm having trouble with the step highlighted in red. Can someone explain the algebra to me? Taken from: Discrete and Combinatorial Mathematics: An ...
2
votes
3answers
188 views

Wheel of Fortune Problem

The Summation formula is $$\sum_{i=1}^ni =\frac{n(n+1)}2$$ How is it that we know the integers $1,2,...36$ appear exactly $3$ times. And why do we multiply the sum by $3$ in the last part of the ...
1
vote
1answer
80 views

How can I divide 30 people into 6 different groups of 5 people in 6 ways so that no two groups share two people?

I have a group of 30 people that I need to divide into 6 groups of 5 people in 6 different ways, however I do not want the same people to be together twice. I already have 5 ways written down, but I ...
6
votes
1answer
47 views

Does $K_{15,15}$ decompose into $K_{5,5}-C_{10}$ and $K_{5,5}-(C_6 \cup C_4)$ subgraphs?

Following on from this question: Q: Does $K_{15,15}$ decompose into $K_{5,5}-C_{10}$ and $K_{5,5}-(C_6 \cup C_4)$ subgraphs? or equivalently Q: Does there exist a $15 \times 15$ matrix ...
-1
votes
1answer
57 views

Quick Truth Table in Logic Problem

Suppose We Have: How can quickly detect how many "1" are in the truth table of above formula? (without drawing Truth Table). i think by using some inference. any idea? we know there are 11 "1"s ...
5
votes
2answers
92 views

Does $K_{12,12}$ decompose into $K_{4,4}-I$ subgraphs?

Q: Does the complete bipartite graph $K_{12,12}$ decompose into $K_{4,4}-I$ subgraphs, where $I$ is a $1$-factor (i.e., a perfect matching)? The obvious necessary conditions work: $K_{12,12}$ ...
0
votes
3answers
81 views

Combination Problem Understanding

How many ways can a Doctor go to the Hospital on $5$ days of January (which has $31$ days) such that no two visits are on consecutive days? I think the solution is: $\displaystyle\binom{27}{5}$ But ...
0
votes
0answers
26 views

How to calculate the number of combinations of getting a pair in a deck of 52 cards?

I am confused over calculating the number of ways in which I can select a pair out of a deck of 52 cards, this is how I go about solving the problem, following the definition of a pair in card games, ...
4
votes
1answer
42 views

All $k$-regular subgraphs of $K_{n,n}$ have a perfect matching: a proof without Hall's Marriage Theorem?

There are several ways of describing this result: Theorem: For $k \in \{1,2,\ldots,n\}$, any $k$-regular subgraph of $K_{n,n}$ has a perfect matching (also known as a $1$-factor). I tend to ...
0
votes
1answer
98 views

Big Mathematics Challenge on Set and Summation? [closed]

please be aware that this is not homework. it's past PHD entrance Exam on 2011. Suppose: $$B=\{(A_1,A_2,A_3) \mid \forall i; 1\le i \le 3; A_i \subseteq \{1,\ldots,20\}\}$$ if we have: ...
0
votes
1answer
17 views

Planner Combination Problem on Graph

I ran into a Graph Problem. Suppose G is A Planner Graph with 100 Vertices such that if connect each two Non-adjacent vertices, the resulting graph would be non-planner. what is the number of edges ...
1
vote
1answer
42 views

Perfect Matching Combination Problem

We know: A perfect matching (a.k.a. 1-factor) is a matching which matches all vertices of the graph. if we remove edges of perfect matching of a 12-Complete Graph. how many triangle remain in this ...
1
vote
3answers
32 views

Problematic Permutation Problem

i see a problem without any definition. would you please help me? i want to calculate the number of permutations of 1,2,...,1392 that 696 numbers be in the natural positions (from all numbers, 696 ...
4
votes
1answer
40 views

Derive a procedure to select one of the 2 options with equal probability when we are not using a fair coin.

Derive a procedure to select one of the 2 options with equal probability when we are not using a fair coin. $P(\text{H}) = p$. $P(\text{T}) = 1 - p = q$. I came up with the following two-roll ...
1
vote
3answers
73 views

Counting valid tickets

I think my question is very easy but I need to understand. The problem is, I have a ticket with 2 numbers from 1 to 10. The first number cannot be greather than the second number. How many valid ...
2
votes
3answers
36 views

let A be a set of 6 distinct postive integers each <= 12, show that the sum of non empty subests of A cannot all be distinct [closed]

let A be a set of 6 distinct postive integers each <= 12, show that the sum of non empty subsets of A cannot all be distinct. for when does this not continue to hold up ( ie instead of 12 , its ...
1
vote
1answer
43 views

Permutations, Combinations, and Counting

A group of 63 people are camping together. They have two 6-person tents, three 4-person tents, five 3-person tents, and three 2 person tents. 18 people will sleep outside of the tents under a tarp. ...
2
votes
1answer
49 views

How many answers can be created using the elementary arithmetic operators?

If I gave you an amount of $n$ numbers, how many anwswer will you be able to create using the elementary arithmetic operators ($+, -, \times, /$)? These are the rules: All numbers ...
0
votes
2answers
69 views

Show that $\binom{n}{k}< \binom{n}{k+1}$ if and only if $k < (n-1)/2$ [closed]

Show that $\binom{n}{k} < \binom{n}{k+1}$ if and only if $k < \frac{n-1}{2}$ and then use this to deduce that the maximum of $\binom{n}{k}$ for $k=0,1,\dots,n$ is $\binom{n}{\lfloor ...
0
votes
0answers
12 views

Must the weight function be nonnegative for the greedy algorithm to be optimal for both a matroid and a greedoid?

Must the weight function be nonnegative for the greedy algorithm to be optimal for both a matroid and a greedoid? For a matroid, the codomain of the weight function is $[0,\infty)$, from Wikipedia ...
3
votes
1answer
31 views

how many ways can 1001 people win 500 identical items?

the question is stating that $1001$ people are in a race and there are $500$ objects that are identical (say the same shirts). We need to find the number of ways that the 500 shirts can be given out ...
1
vote
1answer
43 views

Meaning of the characteristic polynomial of a matroid

From wikipedia The characteristic polynomial of a matroid $M$ (which is sometimes called the chromatic polynomial,[29] although it does not count colorings), is defined to be $$ p_M(\lambda) ...
9
votes
1answer
86 views

How many Sudoku puzzles are there with at least one solution?

A Sudoku puzzle is a 9 by 9 matrix of blanks(which we can represent as 0), and elements of the set {1,2,3,4,5,6,7,8,9}. How many Sudoku puzzles are there with at least one solution. Yes, I am even ...
7
votes
5answers
171 views

Does $\{1,2,\ldots,3000\}$ contain a subset of $2000$ integers with no member twice another?

Does the set $X=\{1,2,\ldots,3000\}$ contain a subset $A$ of $2000$ integers in which no member of $A$ is twice another member of $A$? I started by putting $P=[1501,3000]$, but twice any integer in ...
2
votes
2answers
75 views

Combination small challenge problem

How many ways can 3 different Scientific Groups be formed using 5 students such that Each student is at least be a member of one committee and each two committee has exactly 2 students in common? I ...
2
votes
2answers
56 views

Equivalent definitions for a coloop?

From wikipedia, in a matroid, An element that belongs to no circuit is called a coloop. Equivalently, an element is a coloop if it belongs to every basis. I wonder why the equivalence? From ...
0
votes
1answer
17 views

How do the dependent sets of a matroid characterize the matroid?

Wikipedia says: The dependent sets of a matroid characterize the matroid completely. The collection of dependent sets has simple properties that may be taken as axioms for a matroid. So I ...
1
vote
0answers
28 views

Name for variations of elements from several sets

Consider the set $S=\{1,2,3\}$. As is well known, $(1,1), (1,2), (1,3), (2,1), (2,3), \ldots, (3,3)$ are the variations with repetition of elements of $S$ taken two at a time. We can similarly ...
1
vote
1answer
25 views

What kind of set system is defined to have this property?

Let $E$ be a set, and $F \in \mathcal P(E)$ has the following property: For every $x\in E$ and $Y,Z\in F$ with $x\notin Y\cup Z$, there exists $X\in F$ with $(Y\cap Z)\cup\{x\}\subseteq X$. I wonder ...
0
votes
0answers
18 views

$k$ edge-disjoint $r$-arborescences in an acylic digraph

An $r$-arborescence of a digraph $D$ is a rooted spanning tree with root $r\in V(D)$ in which all the edges of $D$ are directed away from $r$. I would like to prove the following: I have thought ...
0
votes
0answers
46 views

Combination Problem on Create Groups !!!

I'm getting stuck on this combination problem I ran into on a previous final exam: How many ways can 3 different Scientific Groups be formed using 5 students such that each student is at ...
0
votes
1answer
49 views

How many cards of a single suit must be present in any set of n cards?

In a standard deck with 52 cards, 4 suits with 13 cards per suit. I feel like I may be looking at this question wrong from the angle of probabilities. How do I answer this?
1
vote
2answers
49 views

Prove that a complement graph of a tree is either connected or it's a union of an isolated vertex and a full graph

I managed to prove the second part - that a tree that is one vertex with n-1 degree and all the rest are connected to it - the complement graph of such tree is an isolated vertex and the rest of the ...
1
vote
0answers
20 views

A set system generated by a closure operator?

Given a ground set $E$, and a matroid closure operator $\tau$ on $\mathcal P(E)$, we can define a set system $(E,F)$ with $$ F := \{X \in \mathcal P(E): \forall x \in X, x \notin \tau(X-\{x\}) \}$$ ...
2
votes
0answers
25 views

what does “modular” mean?

I find some similarity of the concept "modular set functions" to the cardinality function. But I don't see the cardinality function is also called "modular" or something else. I wonder what "modular" ...
0
votes
2answers
30 views

How many different ways are there to put $9$ coins in $9$ boxes if…

the coins are all identical and exactly six boxes are empty. My first thought was that it should be ${9\choose6} {{3+9-1}\choose9} $, first choosing the $6$ empty boxes and then distributing the ...