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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0answers
20 views

Find number of rectangles

There is NxM grid present with numbering as 1,2,...NM(numbering is done row wise. 1st row will contain number from 1 to Mstrong text, second row will contain M+1 to ...
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1answer
12 views

Turning Preordered Sets into Preordered Monoids (Constructing Preordered Monoids from Preordered Sets)

Question: Referring to the Wikipedia article on Adjoint Functors in Section 2 (Motivation), they talk about "turning rngs into rings" (can be rephrased as "constructing rings from rngs"). I do not ...
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6answers
43 views

Probability of getting $5$ heads on $10$ (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely ...
2
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2answers
37 views

How many bit strings of length 8 begin and end with a 1?

A bit string is a finite sequence of $0$’s and $1$’s. How many bit strings of length $8$ begin and end with a $1$? My answer would be: $2^6$. Because we know, that the bit starts with $1$ and ...
4
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1answer
22 views

What is the probability that these two objects are of the same color?

We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, ...
2
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1answer
27 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
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1answer
16 views

Counting “How many ways to choose courses to graduate” with constraints

I have a problem like, "to graduate you must choose 6 out of 20 courses, but at least 2 out of the 6 courses must be a math course. 8 out of the 20 offered courses are math courses. How many choices ...
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1answer
20 views

show that the maximum degree of the graph is 6

Let p1, p2, . . . , pn be n points in the plane such that the distance between any two points is at least one. Let G = (V, E) be the graph such that V = {p1, p2, . . . , pn} and E = {pipj | distance ...
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2answers
20 views

How many times will the innermost loop be iterated

How many times will the innermost loop be iterated when the algorithm segment is implemented and run? Assume $n$, $m$, $k$, and $j$ are positive integers. ...
5
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2answers
52 views

Minimum elements present in {0, 1, 2, …, 225} to guarantee triple which sums to 225

Suppose I have the set: $$A=\{0, 1, 2, ... 224, 225\}$$ I want to find a triple that sums to $225$ (where a triple is a set of 3 unique values from the set). No Repetition Version: There are many ...
11
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3answers
250 views

Expected value problem with cars on a highway

There is a very long, straight highway with $N$ cars placed somewhere along it, randomly. The highway is only one lane, so the cars can’t pass each other. Each car is going in the same direction, ...
1
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1answer
15 views

5-tuples of n integers

If n is a positive integer, how many 5-tuples of integers from 1 through n can be formed in which the elements of the 5-tuple are written in decreasing order but are not necessarily distinct? In other ...
0
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1answer
32 views

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads I already know that the exact same question has already been answered here But I am trying to solve it ...
2
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1answer
31 views

In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, …, 500$ such that one number is the average of the other two?

Here's the question which I'm struggling with - In how many ways can we pick a group of 3 different numbers from the group $1, 2, 3, ..., 500$ such that one number is the average of the ...
4
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2answers
30 views

Pair of friends and a pair of “enemies” in each group of three students

The problem: There is a class. In each group of three students in the class there is a pair of friends and a pair of "enemies". Find the maximum number of students in the class. I tried to play with ...
0
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1answer
27 views

How many non-congruent triangles with perimeter 11 have integer side lengths? [on hold]

How many non-congruent triangles with perimeter 11 have integer side lengths? I failed to solve it. Can anyone help?
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2answers
24 views

Combinatorics: Does this method take into account every possible matchup

It's actually a question on finding the probability, but I am stuck on a different part of this question. There are $2^n$ players playing a tennis tournament. I have to find the total number of ways ...
0
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1answer
23 views

number of ways of arranging balls so that there are exactly two pairs of green balls

There are $5$ identical red balls and $6$ identical green balls. In how manys we can arrange them so that there are exactly two pairs of green balls. Let red balls be $R,R,R,R,R$ and green be ...
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0answers
42 views

How many ways to pick four increasing numbers from 1 through 39?

Say there's a bin of balls numbered 1 through 39, how many ways are there to pick 4 increasing numbers in a row? First I figured that it would be a permutation without repetition problem, so I got ...
2
votes
2answers
34 views

constrained stars and bars problem

I want to know number of solutions for following equation, where $r_k$'s are non-negative integers, and there is a constraint on $r_k$'s such that $r_1 \geq r_2 \geq \cdots \geq r_K$ \begin{equation} ...
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0answers
15 views

Small graphs containing all trees on $n$ vertices

What do those graphs look like which contain a copy of every tree on $n$ vertices and such that no proper subgraph has this property?
4
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2answers
108 views

What am I counting wrong?

EDIT: I made a mistake in the beginning, the second condition has changed. Sorry for this. I'm asked to count the number of sets of 4 elements that satisfy the two following conditions: 1) Each ...
-1
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0answers
26 views

Number of solutions of equation with natural numbers [on hold]

Given natural numbers $s, n, k$. How to find number of solutions to equation $a_1 + a_2 + \ldots + a_s = n-s$ where $0 \leq a_i \leq k-1$ and $a_i \in \mathbb{N}$?
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0answers
40 views

all but one sub-strings within a cyclic string

over $GF(q)$ where $q\in\mathbb{N}$, we build a string of size $q^n-1$. now, how can I show that it is impossible to construct that string so it contains all sub-strings of size $n$ exactly once, but ...
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0answers
17 views

Onto functions from a set with 4 elements to a set with 3 elements [duplicate]

How many onto functions are there from a set with four elements to a set with three elements? If the four elements set is A = {a, b, c, d} and the three elements set is B = {u, v, x} I see these ...
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0answers
28 views

What is the probaility that two random permutations have same order?

I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one ...
0
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2answers
27 views

equation to create unique value

I have a list of n objects say [ apple, orange, carrot, cherry, banana ] Now I am trying to come up with an equation which will generate an unique number for ...
3
votes
1answer
55 views

Product of sums into a sum of products

Any idea on how I can get an expression in the form of sum of products from the following one?: \begin{equation} \prod_{i=1}^M \left(\sum_{n=1}^i x_n\right) \end{equation}
1
vote
1answer
36 views

The probability of being dealt at least 5 wanted cards

In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards). I want to answer the question: I am dealt 10 cards. I wish to receive 5 ...
5
votes
1answer
86 views

How to solve this hard sum problem?

$$\sum _{ x=1 }^{ \infty }{ \frac { 3{ x }^{ 2 }+12x+16 }{ { \left( x\left( x+1 \right) \left( x+2 \right) \left( x+3 \right) \left( x+4 \right) \right) }^{ 3 } } } =\frac { 1 }{ 4{ (a!) }^{ b } } ...
9
votes
2answers
117 views

Number of ways to partition $40$ balls with $4$ colors into $4$ baskets

Suppose there are $40$ balls with $10$ red, $10$ blue, $10$ green, and $10$ yellow. All balls with the same color are deemed identical. Now all balls are supposed to be put into $4$ identical baskets, ...
6
votes
2answers
86 views

Given the set $A=\{1,2,\dotsc,14\}$, find all subsets of $7$ elements that sum to a multiple of $7$.

I would appreciate if somebody could help me with the following problem. Given the set $A=\{1,2,\dotsc,14\}$, calculate the number of distinct sets $M \subset A$ such that $|M| = 7$ and such that ...
3
votes
2answers
58 views

Find number of ways to seat $n$ boys and $n$ girls in a row so that every boy has atleast one girl sitting beside him.

My attempt: I am getting $2^n(n!)^2$ . First I paired $n$ boys and $n$ girls in $n!$ ways then these pairs can be arranged in $n!$ ways and in each of these pairs boy and girl can arrange themselves ...
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1answer
43 views

Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution. Edit:- ...
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2answers
34 views

Binary Strings: How to determine if decomposition is ambiguous

Let's say I have the following decomposition: $$\{100,10011,00110\}^*$$ How would I determine if the decomposition is ambiguous or unambiguous?
0
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1answer
34 views

More intuitive way for solving this problem than using the multinomial theorem?

I'm the TA in a discrete math course and there was a problem in this weeks problem set which I had troubles solving. It goes like this: Find the coefficients of $v^2w^4xz$ in the expansion of $(3v + ...
0
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0answers
28 views

Using a combinatorical proof for recursion

I am having trouble understanding a combinatorial proof. I have a recursion, $$ a(n) = 2*a(n-1) - a(n-2) $$ And the combinatorial explanation (i.e., proof-light) is that $a(n)$ is just the count ...
1
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1answer
22 views

Need combinatorial formula

Let we have a forest $F_n(P)$ with $n$ nodes defined by set $P$ of all pairs $\{\text{father}, \text{son}\}$. For instance $P=\{\{1, 2\}, \{3, 4 \}, \{1, 3 \}\}$ defines a forest $F_5(P).$ Let ...
0
votes
1answer
23 views

Probability of drawing $m$ of $A$ in $n$ cards given a deck of $d$ cards contain $a$ copies of $A$?

As in the title I'm trying to work out what the chances of drawing $m$ copies of a specific card in $n$ draws are given a deck size of $d$ containing $a$ copies of $A$. I've tried using permutations ...
2
votes
2answers
33 views

Counting permutations with given condition

I need to find number of permutations $p$ of set $\lbrace 1,2,3, \ldots, n \rbrace$ such for all $i$ $p_{i+1} \neq p_i + 1$. I think that inclusion-exclusion principle would be useful. Let $A_k$ be ...
0
votes
1answer
34 views

In how many different ways can the gifts be given? [on hold]

For Valentine's Day $5$ children receive a total of $6$ different gifts. Each child receives at least one gift and each gift is given to exactly one child. In how many different ways can the gifts ...
1
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1answer
29 views

Show impossibility of a perfect covering

Problem: Show that a $8 \times 8$ chessboard cannot be perfectly covered by $1$ square tetramino, and 15 other tetraminoes chosen from straight tetraminoes and Z-tetraminoes. My attempt: I tried to ...
3
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1answer
21 views

How many teams can be formed?

I would like to calculate the number of choices of teams I can make in the following scenario. Suppose a team is comprised of 3 characters (1 leader and 2 support members) and suppose there are 108 ...
5
votes
3answers
38 views

Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$

I need to show that $$\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$$ I know that $\displaystyle \binom{n}{k}\binom{k}{i}$ is counting the number of ways to pick $k$ elements ...
2
votes
1answer
35 views

Probability of an array having all distinct numbers

Suppose you have an array of size $2n$. There are two times $2n^2$ distinct numbers that can be put into the array without replacement, i.e. for each choice of number, there are two copies, so a ...
1
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1answer
18 views

Number of permutations of $S_n$ such that $\sigma^h(a) = \sigma^k(b)$

A basic result in combinatorics is: In $S_n$ there are $(n-d)(n-2)!$ permutations $\sigma$ such that $\sigma^k(a) = b$, if $a \neq b$; $d(n-1)!$ permutations $\sigma$ such that ...
1
vote
1answer
20 views

How many options are there to award gold, silver, and bronze medals to a group of $10$ athletes?

How many options are there to award gold, silver, and bronze medals to a group of $10$ athletes? Is this permutation or combination, and is there repetition? I thought this would be a combination ...
1
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0answers
42 views

Estimates for the Dedekind number $M(9)$

The Dedekind number $M(n)$ is the number of antichains in the partial order of subsets of $\{1,\dotsc,n\}$. It is only known for $0 \leq n \leq 8$. Question. What are some known upper and lower ...
5
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1answer
45 views

Number of $n$-digit permutations with exactly $n-2$ digits smaller than the next

How many permutations of $1,2,\cdots, n$ contain exactly $n-2$ digits that are smaller than the digit immediately to their right? My solution proceeded with recursion. It has some chance of being ...
1
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0answers
26 views

Possible closed form or approximation?

Does it have some closed form or approximation ? I tried on my own but i am not getting any idea regarding this. $$\sum_{k_1=k}^{M}\sum_{k_2=k}^{M}\frac{k_1^{-\gamma} k_2^{-\gamma} ...