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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13 views

Confused between cyclic sum and symmetric sums.

four variables $a, b, c, d$ are given, what is the symmetric and cyclic sum? I thought: $$\sum_{cyc} ab = ab + ac + ad + bc + bd + cd$$ And $$\sum_{sym} ab = 2(ab + ac + ad + bc + bc + ...
3
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2answers
36 views

Number of ways to choose numbers from a list.

While studying, I came upon this question in my book: "How many ways are there to take 7 numbers from 1 to 12 such that none of the chosen numbers is twice the other?" The solution is shown as 47, but ...
1
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2answers
45 views

A bank has to give 5 positions for 15 candidates

A bank must give 5 different positions to 15 people: 7 men and 8 women. Question 1: In how many ways can the jobs be given if there must be at least 3 women selected? Question 2: In ...
4
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0answers
21 views

Show that $p \in \left[\frac{4^m}{\sqrt{2m}},\frac{4^m}{\sqrt{2m+1}}\right]$

If the number of ways in which $m$ identical apples can be put in $2m$ boxes, so that no box contains more than one apple, is $p$, prove that $$p \in ...
2
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0answers
31 views

Graph theory, $n$ people sitting around table.

$n$ people want to have dinner together around a table for $k$ nights so that no person has the same neighbor twice. How big can $k$ be in terms of $n$? Does everybody get to sit next to everybody ...
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2answers
37 views

How To Approach Dice Rolls

When asked about 2 dice roll, we do we count the result that both dice have the same number just one time and not two? If it is because we can distinguish between the two, so if the dice was colored ...
0
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2answers
29 views

Comparison of two sets of 4-tuples using combinatorics

My problem is to show that $\mathbf{A} = \mathbf{B}$. Specifically that $\forall a \in \mathbf{A} \implies a \in \mathbf{B}$ and $\forall b \in \mathbf{B} \implies b \in \mathbf{A}$, to be precise. ...
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1answer
26 views

No of ways of selecting r objects from n distinct objects, allowing repeated selections

I'm self studying discrete math from a books which states the formula for No of ways of selecting r objects from n distinct objects, allowing repeated selections as $C(n+r-1, r)$. I couldn't ...
2
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0answers
19 views

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2n−1$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$.

Maximal determinant of a $\{1,−1\}$ matrix of size $n$ is $2n−1$ times the maximal determinant of a $ \{0,1\}$ matrix of size $n−1$. How to prove this result? (I found this statement while reading ...
2
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2answers
81 views

${2000\choose1}+{2000\choose4}+{2000\choose7}+\cdots +{2000\choose1996}+{2000\choose1999}=?$ [on hold]

${2000\choose1}+{2000\choose4}+{2000\choose7}+\cdots +{2000\choose1996}+{2000\choose1999}=?$
-2
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1answer
33 views

Calculate $\left(\begin{smallmatrix}n \\ r\end{smallmatrix}\right)/{k^n}$ for very large $n$

How to calculate large $ \frac{\left(\begin{matrix}n \\ r\end{matrix}\right)}{k^n}$, given very large $n$. Since n is large enough normal methods of calculating $ \left(\begin{matrix}n \\ ...
2
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4answers
78 views

A combinatorial proof for $\binom mk$+$\binom m{k-1}$=$\binom {m+1}k$

I do realize that there is a elementary proof of this result which follows from applying the formula $$\binom mk=\frac{m \cdot (m-1) \cdot \ldots \cdot (m-k+1)}{k!}.$$ I do wonder if there is an ...
1
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2answers
40 views

Another kind of derangement?

I reading about derangements, and the following question came to my mind. Suppose in an office, there work 5 teams, each consisting of 1 head and 3 staff (so there is a total of 15 staff). If the ...
-1
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1answer
32 views

How many different three-digit house numbers could be made?

a shopkeeper sells house numbers. she has a large supply of the numerals 4, 7 and 8, but no other numerals. how many different three-digit house numbers could be made using only the numerals in her ...
0
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2answers
50 views

3 cards are drawn from a deck of 52. how many hands are possible if exactly 2 are black cards and exactly 1 is an ace? [on hold]

3 cards are drawn from a deck of 52. how many hands are possible if exactly 2 are black cards and exactly 1 is an ace? I'm not sure how this works. Never seen a question quite like this.
2
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3answers
212 views

Stumped - How would I solve this probability question?

This question was merely a fun online math problem to see how many people could solve it, but I haven't been able to since last week and it's beginning to drive me nuts. The question: A man has $7$ ...
4
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0answers
62 views

Algorithm to find shortest path to net values across nodes

I have an undirected graph $G = (V, E)$ with nodes $V$ and edges $E$. Each node $v$ has an associated number $n_v \in \mathbf{Z}$ Let us define for any two nodes $v, w \in V$ connected by an edge $e ...
4
votes
3answers
254 views

Proof for coloring combinations problem. (color vertices of pentagon)

While studying, I found a problem in my book that read: "Each vertex of convex pentagon ABCDE is to be colored with one of seven colors. Each end of every diagonal must have different colors. Find the ...
1
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1answer
41 views

Maths challenge problem: Why is the number of teams which require 4 substitutions 32?

I came across the following problem on a UKMT senior maths challenege: A hockey team consists of 1 goalkeeper, 4 defenders, 4 midfielders and 2 forwards. There are four substitutes: 1 goalkeeper, 1 ...
2
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1answer
21 views

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x + y < n$.

Let $n$ be a positive integer and $S$ the set of points $(x,y)$ in the plane, where $x$ and $y$ are non-negative integers such that $x + y < n$. The points of $S$ are colored in red and blue so ...
8
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2answers
48 views

How many truth tables if there are only $\land$ or $\lor$ for $n$ variables?

For example, if we have three operators $\land, \lor$ and $\neg$. For $n$ variables, there will be $2^{2^n}$ different truth tables. Because for $2^n$ rows of the truth table, there are $2$ choices - ...
0
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1answer
14 views

Expectation of size of bootstrapped sample

Lets say we have a sample $\mathbf{X} = \{x_1, x_2, \dots, x_N\}$. We draw $N$ points from $\mathbf{X}$ with replacement (do a $\textit{bootstrap})$. What is the expectation of size of bootstrapped ...
3
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2answers
38 views

A grasshopper starts at the origin and is equally likely to hop north,s,e,w. What is the probability that it's coordinates will be 0,0 after 4 hops?

The grasshopper must hop in all $4$ directions (North, South, East, and West) to get back to the origin after $4$ hops. Therefore, I did: $\frac{(4 \cdot 3 \cdot 2 \cdot1)}{4^4} = .09375$. However, ...
5
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0answers
37 views

Given $100$ coplanar points, no $3$ collinear, then at most $70$ percent triangles formed using these points are acute-angled

(IMO-$1970$) Given $100$ coplanar points, no $3$ collinear, prove that at most $70$ percent of the triangles formed using these points are acute-angled. I know that one solution proceeds by ...
4
votes
2answers
31 views

Number of ways to select subsets

In how many ways can two distinct subsets of the set $\text{A}$ of $k$ $(k \geq 3)$ elements be selected so that they have exactly two common elements? I started by choosing two elements (that ...
0
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2answers
62 views

Powerset with constraints

I have two sets $NUMBERS$ and $LETTERS$ with: $ NUMBERS = \{1, 2, 3, 4, 5\} \\ LETTERS = \{ A, B, C, D, E\}$ No I want the power-set of my sets, i.e. the set of subsets of elements from both ...
2
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0answers
50 views

Number of games required such that two arbitrary players play together and against each at least once.

There are $2N$ players to form two teams of $N$ players that play against each other in a game. How many games are required such that two arbitrary players play together and against each other at ...
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2answers
49 views

How to solve this combinations with repetitions problem using generating functions?

Find the number of solutions to : $$x_1 + x_2 + x_3 + x_4 + x_5 = 10$$ where none of the variables can be the number $3$. I can solve this with Inclusion-Exclusion Principle, but I really love ...
0
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0answers
12 views

Optimization problems with combinations of a finite set as the feasible area?

For example: Provided that $S\subset \Re$ is a known finite set ($n\leq |S| < \infty$), number $k$ is known, and $1 \leq k<n$ minimize $f(x_{1},\ldots, x_{n}) = \sin (\sum_{1\leq i\leq ...
3
votes
1answer
111 views

Expected value when die is rolled $N$ times

Suppose we have a die with $K$ faces with numbers from 1 to $K$ written on it, and integers $L$ and $F$ ($0 < L \leq K$). We roll it $N$ times. Let $a_i$ be the number of times (out of the $N$ ...
0
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0answers
16 views

subsets with predefined sequences

I have a set $N=\{m,m+1,m+2,...,n\}$ And there are some generating functions of the format : $f(x,k) = (x^2 -1) \mod k$, where $k \le \sqrt m$ and $k$ is in the form $(6i+1)$ or $(6i-1)$, $\forall ...
0
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1answer
20 views

permutations vs combinations on slot machines with repeating elements on each reel

For a slot machine with 5 reels where there are repeated elements on each of the reel. Example: Reel 1 [ 1, 1, 2, 1, 3, 5, 6 ] Reel 2 [ 1, 2, 3, 4, 5, 5 ] Reel 3 [ 2, 2, 3, 2, 4 ] Reel 4 [ 1, 2, 3, ...
3
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1answer
67 views

Interesting Combinatorial Identities; e.g. $\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$ [duplicate]

I came across the following combinatorial identity: $$\sum_{k=0}^n {n\choose k}^2 = {2n\choose n}$$ Here's the kind of proof which caught my interest: $\sum_k {n \choose k}^2 = \sum_k {n \choose ...
-7
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0answers
37 views

Fundamental principle of counting? [on hold]

How many three-digit even numbers are there such that 9 comes as a succeeding digit in any number only when 7 is the preceding digit and 7 is the preceding digit only when 9 is the succeeding digit? ...
0
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1answer
49 views

Probability a blackjack dealer will bust if you know their score and know the exact deck?

If you know the exact cards left in a deck, and the score of the dealer, how can you calculate the exact probability that they will bust? The dealer behaves as follows: If the dealer's score is less ...
1
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1answer
38 views

How many surjective functions $f: X \to \{1,…,j\}$?

How many surjective functions $f: X \to \{1,...,j\}, |X|=j \cdot k.$ can be defined if they must satisfy: $$ |\{x\in X: f(x)=r\}|=|\{x\in X: f(x)=s\} \forall r,s\in \{1,...,j\} $$ My attempt: From ...
1
vote
1answer
40 views

Number of distinct necklaces using K colors

I have a task to find the number of distinct necklaces using K colors. Two necklaces are considered to be distinct if one of the necklaces cannot be obtained from the second necklace by rotating ...
1
vote
1answer
29 views

maximal matching in graph theory

if we have a graph $G = (V,E)$ and the four values $\beta_1(G)$, $\alpha_1(G)$, $\beta(G)$, $\alpha(G)$, where $\beta_1(G)$: Edge independenth number. The maximal number of independent edges in the ...
5
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2answers
118 views

Determine the number of subsets

How many distinct subsets of a set $\text{A}$ are there, containing at least $9$ elements, where the total number of elements in set $\text{A}$ is $18$ ? I've solved it by making cases of either ...
0
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0answers
16 views

Knight paths on homothetic polyominoes

A while back I made the following conjecture : Let $P$ be an arbitrary polyomino .Let a polyomino be good if there exist a path of a knight on it which passes through each little square exactly once ...
0
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1answer
52 views

How to interpret the Generalized Version of Inclusion-Exclusion Principle

This is a follow-up question on the previous post. Let's say there are $n$ properties which are numbered $1,\cdots,n$. And let $A$ be a set of elements which has some of these properties. Then the ...
1
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1answer
25 views

The number of ways to draw boundaries of constituencies, subject to constraints

A state comprises 45 counties arranged as 5 rows, running east and west, of 9 counties each, the nine colums of 5 running north and south. They're connected horizontally and vertically, i.e. ...
1
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1answer
18 views

Numbers written into a square grid

I was working on a problem from The Art and Craft of Problem Solving by Zietz, in the chapter called 'The extreme principle.' Here is the problem: "The integers from 1 to $n^2$ are written into a ...
2
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2answers
25 views

Special case on counting in a string of 7 letters

I have the following question: Suppose $S_7$ is the set of all strings of length seven that can be formed with the letters $A, B, C, D, E, F$ and $G$ when repetitions are allowed. How many strings ...
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0answers
47 views

Recall structures made from legos [on hold]

Recall structures made from legos. We do not see these as just one lego brick after another, we see substructure. Try to find some substructure in the following lines of proof. Assume r is in Q. ...
4
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1answer
23 views

Arrangement of any number of objects from $n$ objects

Prove that the total number of arrangements of objects by taking any number of objects from $n$ different objects is $\lfloor e \times n! - 1 \rfloor$, where $e$ is the natural base. I tried it ...
1
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1answer
34 views

Is this a binomial or multinomial question?

You can donate to a company: $10$ dollars , $20$ dollars or nothing. In a mall there are $70$% young people and $30$ % old people. $50$% from the old people aren't donating anything. ...
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0answers
22 views

How many ways shuffle $n_1$ and $n_2$ balls when we but them together?

I have $n_1$ white balls and $n_2$ black balls, and I want to know how many ways I can make a distinct arrangement from them. For example , $n_1 = 2$, $n_2 = 1$ then there are three distinct ...
0
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1answer
19 views

Counting weakly connected graphs with outdegree of exactly one.

If we count all graphs of $N$ labelled vertices, where each vertex has an outdegree of exactly $1$ with no self-loops allowed, we'll find that there are exactly $(N-1)^N$ of them (for every of $N$ ...
2
votes
2answers
40 views

Erin rolls 4 four-sided dice all at once, then can roll a subset of her choosing a 2nd time. What is the probability of getting all the same number?

Here's what I have so far: All 4 same on first try = (1/4)^4 * 4 3 same, then get 4th on 2nd roll = 4 * (1/4)^3 * (3/4) * (4!/3!) Here's where I'm confused: 2 same = 4 * (1/4)^2 * (3/4)(2/4 :to ...