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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2
votes
4answers
56 views

Given that $6$ men and $6$ women are divided into pairs, what is the probability that none of the women will sit with a man?

I've generalized the question I was given here for simplicity: $6$ men and $6$ women are to be paired for a bus trip. If the pairings are done randomly, what's the probability that no women will end ...
-2
votes
0answers
15 views

Confusing Conditional Probability question 68 [on hold]

The four top tennis players in the world A, B, C, and D are invited to a special tournament where the winner gets one million dollars. In round one, Player A plays player D and player B plays player ...
2
votes
4answers
54 views

Prove the formula $\sum_{k=1}^n k\binom{n}{k} = n \cdot 2^{n-1}$ for all integers $n > 0$ [duplicate]

I just got to this question and I became a question mark. I wonder if anyone can help me with this one, because I don't even know how to begin to tackle this problem. The question: Prove the ...
2
votes
1answer
46 views

In how many ways can $8$ appointments be scheduled for a physician visiting a nursing home with $20$ patients? [on hold]

A physician routinely visits a local nursing home on Thursday mornings to examine patients. Suppose the facility has $20$ residents, but the physician only has time to check $8$. The supervisor places ...
0
votes
0answers
3 views

Counting subgraphs of bounded extremal degrees

Let $m\leq n-1$. Is there a closed expression counting the subgraphs of minimum degree $\geq m$ (resp. maximum degree $\geq m$) on $n$ labelled vertices?
0
votes
1answer
40 views

Finding a closed formula for: $1\cdot2\cdot3+2\cdot3\cdot4+…+(n-2)\cdot(n-1)\cdot(n)$ [duplicate]

As I calculated the sum of the serie above doesn't exist(sum doesn't converge). How can I prove it using the double computing(combinatorical method)?
1
vote
1answer
26 views

How many ways to divide $n$ different pieces of chocolate in two non empty groups?

After the example I think that the order of the groups doesn't matter so ${(A),(B,C)}$ and $(B,C),(A)$ counted as $1$. Suppose we split $5$ chocolates into a group of size $1$ and a group of size of ...
-1
votes
5answers
68 views

Deck of Cards Stats Probability Question [on hold]

Randomly select two cards in sequence from a full deck of 52 cards, what i s the probability that the first one is a King given that the second one is a King. If someone can please help me with this ...
2
votes
0answers
26 views

Distribution of distinct object problem

So i was given this question. How many ways are there to place 10 distinct people within 3 distinct rooms with exactly 5 people in the first room and 2 people in the second room? So i asked this ...
0
votes
1answer
16 views

Summation of all j-combinations (Expanding composition formula)

I found a formula for a problem that I was trying to solve, the Formula 3.2 in Section 3 at page 441 of this document.I am a little unsure about the "Summation over all j-combinations". Here is what I ...
2
votes
0answers
27 views

Generating subsets with 1 common element

I have a number $n$ and a set $S$ of $n(n-1)/2$ elements : $ \{1, 2, \ldots, n(n-1)/2\}$ I'm looking for an algorithm to generate $n$ distinct subsets of $S$, each having $n-1$ elements, with the ...
2
votes
1answer
19 views

How do you calculate the width of the Poset Lattice of Divisors?

Let $n = 10800 = 2^43^35^2$ I can find a set of eleven divisors of $n$ such that none divides another: $$\begin{array}{ccccc} & & & 2 3^3 & 3^35\\ & & 2^23^2 & ...
4
votes
3answers
212 views

How many solutions for equation with simple restrictions

I'm working on an assignment in which I have to count the number of solutions for this particular equation: $$x_1+x_2+x_3+x_4=20$$for non negative integers with $x_1<8 $ and $x_2<6$ I'm aware ...
3
votes
1answer
54 views

Number of 'walks' which stay above 0.

Consider a set of distinct $n$ numbers where $a_i \in \mathbb{R} $ and $$\sum_{i=1}^{n} a_i = 0$$ A walk is defined to be the sum of the numbers, so that the $k$th position is the partial sum to $k$. ...
0
votes
2answers
44 views

How many 10-digit decimal sequences (using 0, 1, 2, . . . , 9) are there in which digits 3, 4, 5, 6 all appear?

So i was given this question. How many 10-digit decimal sequences (using 0, 1, 2, . . . , 9) are there in which digits 3, 4, 5, 6 all appear? My solution below (not sure if correct) Let $A_i$ = set ...
5
votes
2answers
30 views

solve for variable in combination

i have the combination ${n\choose 11}=12376$ and am looking to solve for $n$. it turns out to be $17$. of course can use brute force approach where just plug numbers in for $n$ but am looking for a ...
-1
votes
2answers
36 views

What is the probability that when a deck of cards is shuffled and dealt, exactly 3 of the 4 aces will be dealt within the last 20 cards? [on hold]

I am trying to figure out this problem, I think that it is a "permutations with repetition" type of question.
1
vote
2answers
33 views

Number of ways of selecting 3 numbers from $\{1,2,3,\cdots,3n\}$ such that the sum is divisible by 3

Find the Number of ways of selecting 3 numbers from $\{1,2,3,\cdots,3n\}$ such that the sum is divisible by 3. (Numbers are selected without replacement). I made a list like this: The sum of ...
1
vote
1answer
31 views

Give a recursion for the number h(n) of strings in S of length n.

Let S be the set of strings on the alphabet {0,1,2,3} that do not contain 12 or 20 as a substring. Solving this I got: $$ h(n) = 4h(n-1) - 2h(n-2)$$ with $h(0) = 1, h(1) = 4,h(2) = 14 $. When I did ...
3
votes
1answer
24 views

Probability: Finding the Number of Pears Given Two Scenarios

You have a bag containing 20 apples, 10 oranges, and an unknown number of pears. If the probability that you select 2 apples and 2 oranges is equal to the probability that you select 1 apple, 1 ...
1
vote
1answer
47 views

How many ways are there to place 10 distinct people within 3 distinct rooms with exactly 5 people in the first room and 2 people in the second room?

So I was given this question. How many ways are there to place $10$ distinct people within $3$ distinct rooms with exactly $5$ people in the first room and $2$ people in the second room? I have ...
0
votes
0answers
26 views

Looking for mathematical/combinatorial and computational explanation regarding adding values in a $5 \times 4$ (matrix?) with a constraint.

Given the following matrix (not sure if I should call it that): Matrix $5 \times 4$ I want to add all possible combinations of values such that each Horse gets but one value from each Bookie. What I ...
-1
votes
2answers
49 views

Number of words of length $n$ on the alphabet $a,b,c$ recurrence. [on hold]

Let $a_{n}$ be the number of words of length $n$ on the alphabet $a,b,c$ such that $b,c$ are not adjacent. What is the recurrence relation for $a_{n}$.
1
vote
0answers
21 views

Small tree containing smaller trees

Given $n$, what is the smallest number $N=N(n)$ with the property that there exists a tree on $N$ (unlabelled) vertices that contains a copy of every tree on $n$ vertices? That such $N$ must exist is ...
1
vote
2answers
79 views

Find the recursive definition for the number of strings on 0, 1, 2 avoiding the substring 012?

This is the question $a(n)$ the number of strings on $0, 1, 2$ avoiding the substring $012$ and the answer is $$a(n)=3a(n−1)−a(n−3)$$ with $$a(0)=1,a(1)=3,a(2)=9$$ My question is how to you get this ...
2
votes
3answers
54 views

Abstract Combinatorics

In a library there is a sequence of $n$ books. There is someone that never wants to take books that are neighborhoods of each other. How many possibilities are there, for him, to take $k\le n$ books? ...
1
vote
3answers
62 views

How many different integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 21$ with restrictions

So i was Given this question. How many different integer solutions are there to the equation $x_1 + x_2 + x_3 + x_4 = 21$ $0 \leq x_i \leq 9$? I just assumed it would be ${21+4-4-1 \choose ...
2
votes
2answers
20 views

Why use C(n,r) instead of P(n,r) when considering how many strings can be formed in which a specific letter appears before another specific letter?

I am dealing with a problem in which I must determine how many strings can be formed by ordering the letters ABCDE subject to the conditions given. The condition that I am given is that A appears ...
3
votes
0answers
23 views

Extremal set theory problem concerning translations of a set of integers

Let $A$ be a subset of $B = \{1, 2,\ldots,n\}$. Suppose that $F$ is a family of subsets of $B$, each of which is a translation of $A$ and no two of which intersect more than once. What is the maximum ...
0
votes
2answers
43 views

Combinatorics president and votes

There are 5 candidates for presidency and 11 people that can vote at most one of them (so they can decide not to vote). How many combinations of votes are there if no candidate can recieve more than 5 ...
0
votes
1answer
20 views

Expanding Restricted Compositions formula

I recently started to look into restricted compositions and I found a formula for a problem that I was trying to solve, the Formula E at page 441 of this document. In my case I have n =8, k=3, t=1 ...
1
vote
2answers
49 views

Proof $\dbinom{n}{0} - \frac{1}{2}\dbinom{n}{1} + \cdots + (-1)^n\frac{1}{2^{n-1}}\dbinom{n}{n-1}$

For all $n \ge 1$, $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} + \frac{1}{2^2}\binom{n}{2} - \frac{1}{2^3}\binom{n}{3} + \cdots + (-1)^n\frac{1}{2^{n-1}}\binom{n}{n-1} = 0,$$ if $n$ is even               ...
1
vote
0answers
6 views

Number of translated cubes covering a given hypercube in $\mathbb{R}^n$

Let $\Omega \subset \mathbb{R}^n$ be open and bounded, and let $Q \subset \Omega$ be a hypercube. Furthermore, denote by $D$ the $n$-dimensional unit cube $(0,1)^n$. Let $k \in \mathbb{N}$ be big, ...
-5
votes
2answers
62 views

Given any 40 people, at least four of them were born in the same month of the year [on hold]

Given any 40 people, at least four of them were born in the same month of the year. Why is this true?
1
vote
5answers
62 views

Proof $x$, $1+nx≤ (1+x)^n$ [on hold]

Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$. Don't have a idea to start this one. I don't know how to use math induction yet, ...
2
votes
0answers
61 views

Counting integers from $1$ to $n$ with an odd number of divisors in {1,2,3,…,k}

Question Given $n,k$ find the number of integers between $1$ and $n$ that have odd number of divisors in {1,2,3,...,k} Example If $n=10$ and $k=3$, the numbers $1(1),5(1),6(1,2,3),7(1)$ have odd ...
1
vote
0answers
22 views

references of discrete association scheme

I tried to find a book or paper to understanding discrete association scheme but I could not get any book for that. What is the good references for that?
2
votes
2answers
40 views

What is the probability that a psychic correctly “predicts” the outcome of at least 5 out of 10 coin flips?

Assume the psychic is actually just randomly guessing on each flip. The attempt: let E be the event in question number of outcomes per flip = 2 chance of correctly guessing the correct outcome = ...
0
votes
0answers
72 views

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? [on hold]

How many ways are there to choose one-half dozen donuts from $9$ varieties so that there are exactly $4$ glazed? How should I approach this problem? Okay I think it's C(10, 2) because I already have ...
0
votes
0answers
33 views

Combinatorics: number of ways to choose $n$ distinct items from k boxes, each containing $s_i$ items?

Say there're $k$ boxes, each containing $s_1, s_2, s_3, \ldots, s_k$ objects; every object is distinct from another. I want to choose $n$ ($n \leq k$) objects, each from a different box (i.e. no two ...
0
votes
0answers
26 views

HW - Number of subspaces T of a vector space K containing a fixed subspace M.

Given a vector space $K$ of dimension $k$ over a finite field $\mathbb{F}_q$, what is the number of subspaces $T$ of dimension $t<k$ that contain a given subspace $M$ of dimension $m<t$? ...
1
vote
1answer
39 views

Number of ways to choose $k$ subsets such that $ B_1 \cap B_2 \cap \cdot \cdot \cdot \cap B_k = \emptyset$.

Let $ \space n,k \in \mathbb Z \space $ such that $1 \le k \le n \space$. Let $\space A=\{1,2,...,n\}$. Find the number of ways to choose $k$ subsets $\space B_1,B_2,...,B_k\space $ of $A$ such that $ ...
-2
votes
0answers
33 views

Different Ids on mars [on hold]

I am doing some exam questions - and I don't know the answer, can u show How to calculate it and what the answer is? The question: On Mars, each Martian alien, has an ID card with a unique ...
0
votes
1answer
22 views

How many 13-card hands have at least one Jack, King, Queen, or Ace?

So with this question, I came to this math: I have a J, Q, K, and A in four suits, and after having one of those face cards in a hand, now we are left to choose 12 more cards. so then I figure we get ...
0
votes
1answer
18 views

How many mappings are there between these two graphs?

Let $P_{20}$ be a path of length 20 like so: $x_0$-$x_1$-$~\cdots~$-$x_{20}$ and $G$ a cycle of order 3. Allegedly there are $3 \cdot 2^{20}$ mappings $P_{20}\rightarrow G$, which I don't quite see. ...
0
votes
1answer
38 views

Alternate proof to number of monomials in a given degree - “more” rigorous, formal [duplicate]

Let $s$ be the number of variables and $n$ be the degree of the monomials we want to count in $R[X_1,\dots,X_s]$. Then show, that the count is $$\delta(n,s):=\binom{s-1+n}{s-1}.$$ The question ...
1
vote
1answer
37 views

How many ways can the team be created?

I am doing some old exam questions - and I don't know the answer, can some one calculate the result and show how you did it?
0
votes
0answers
9 views

Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
-2
votes
3answers
47 views

Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$

I am given this: Find the number of elements in $A \cup B \cup C$ if there are 50 elements in $A$, 500 in $B$, and 5,000 in $C$ if: $A \subseteq B$ and $B \subseteq C$ The sets are pairwise ...
2
votes
1answer
53 views

Euler and Bernoulli Polynomial Identity Proof

Given that the Euler Polynomials $E_n(z)$ are defined in terms of the generating function $$\frac{2e^{xz}}{e^x+1}=\sum_{n=0}^\infty E_n(z)\frac{x^n}{n!}$$ and that the Bernoulli Polynomials $B_n(z)$ ...