combinatorics's questions - Chinese 1answer

33.156 combinatorics questions.

An urn problem An urn $C$ contains $c\in \mathbb{N}$ elements of three different kinds: There are $\alpha\in\mathbb{N}$ elements of kind $A$, $\beta\in\mathbb{N}$ elements of kind $B$, and $\gamma\in\...

What is the number of ways to construct a sequence of length 8 from 6 differnt numbers from 1 to 10. The sequence must contain all 6 chosen numbers Thank you!

I want to prove the following equation but can't find a good combinatoric proof $\sum_{k=0}^{n}{\binom{2n+1}{k}}2^{n-k} = \sum_{k=0}^{n}{\binom{n+k}{k}}3^{n-k}$

There are 10 languages spoken at a conference, and 10 translators. Each translator speaks exactly two of the 10 languages, with an equal probability of each language. Some translators may speak the ...

I'm not understanding the method of using multinomial theorem in combinatorics problems. For example, suppose we want to distribute $17$ identical oranges among $4$ children such that each child gets ...

In how many ways can I distribute $6$ identical cookies and $6$ identical candies to $4$ children, if each child must receive at least $1$ of each type of item? I know how to distribute the things if ...

Mila has four ropes. She chooses two of the eight loose ends at random (possibly from the same rope) and ties them together, leaving six loose ends. She again chooses two of these six ends at random ...

Let $\mathbb{N}=\{0,1,2,\ldots\}$. Does there exist a bijection $f\colon\mathbb{N}\to\mathbb{N}$ such that $f(0)=0$ and $|f(n)-f(n-1)|=n$ for all $n\geq1$? For example, you might start with $$\begin{...

Recently, I was running through some complementary counting and casework problems when I found this one. "Let's say you have 5 fair six sided dice. You roll all of them. What's the probability that ...

In how many rotational distinct ways can we color the vertices of a cube with 2 colors and faces with 4 colors? (This can be interpreted in two ways, either you have to use exactly 4 colors or at most ...

The $\eta$-value of an integer partition $\lambda = \big( \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_k \geq 0 \big)$ is defined as \begin{equation} \eta \big( \lambda \big) \ := \ \sum_{i=1}^k ...

Consider a general bit string of length $n$; how many bit strings are there that contain a substring $T$? For example, given a bit string of length 6, how many are there that contain $110$ as a ...

There is a small but well-known open problem in graph coloring called the Total Coloring Conjecture. I worked on this problem several years ago and am interested in returning to it, potentially. For ...

Let $N$ be a positive integer. Then, we have $\sum\limits_{j=1}^{N} \binom{j}{6} = \binom{N+1}{7}$. Could anyone explain this equation a little bit? I wrote out the left hand side as $\binom{1}{6} + ...

Setup: I have an undirected (multi)graph $G$ with $n+1$ vertices. Each vertex has degree $\leq 2$, and vertex number $0$ has degree exactly one. What I'm looking for: I know that any graph with ...

I'm teaching an undergrad course in graph theory and have just finished the proof of Max Flow/Min Cut. So far I have used Diestel's definition (more or less) of flow network as a digraph $G$ with a ...

What I'm looking for is the name of a type of number set. Given a number T (for total) and a set of positive integers S, I want to uniquely identify the subset of S that sums to T. All sets containing ...

Problem related to series of binomial coefficients in which each term is a product of two binomial coefficients. In this question: Prove that $$\binom{n}0^2+\binom{n}1^2+\ldots+\binom{n}n^2=\frac{...

For a positive integer $n$, let $A$ and $B$ be the families of $n$-element subsets of $S=\{1,2,\ldots ,2n\}$ with respectively even and odd sums of elements. Compute $|A|-|B|$. If we mark $E = \{2,4,....

I am having difficulty solving this. I know that 6 letter words possible with abacus is (6! / 2! ). But my doubt is what happens when we have to choose 3 letters from abacus and how will we handle the ...

Suppose that we are given a sequences of $2N$ 'entities' (not numbers) with some total ordering defined among these entities. An example could be $$\langle a\rangle=1<4<8<2<3<\cdots<...

Suppose there are two different coupons and we want to find the average number of coupons drawn until we get the two coupons. The probabilities of getting the two coupon types are not equal. Denote ...

Let $A$ be an arbitrary list with $n$ distinct elements. Suppose I have a separate list, $B$, which is initially identical to $A$. Denote individual elements in $B$ as $b_k$ where $k\in\{1,2,...,n\}$...

In a paper by MELVYN B. NATHANSON (https://arxiv.org/abs/1401.7598), it was claimed that: The Goldbach conjecture implies that the set of primes is an asymptotic basis of order 3. My question is: How ...

I have a sequence of n non-distinct numbers and I want to find the number of combinations of half the size. For example: Given 0.5, 0.5, 1, 1 there are 3 combinations of size 2: 0.5, 0.5 1, 1 1, 0.5 ...

Well, this question has different solution depending on whether the toys, balls and rings are identical among themselves or not. My simple query is that by default whether we should consider the toys, ...

Suppose we want to prove $$ k \binom{n}{k} = n \binom{n-1}{k-1}$$ In the LHS we are choosing a team of $k$ players from $n$ players. Then we are choosing a captain. In the RHS we are choosing a ...

Question Let $b_{n,r}$ be the number words of length $r$ over $[n]=\{1,2,\dotsc, n\}$ with no three consecutive letters the same. Show that $$ b_{n,r}=(n-1)(b_{n,r-1}+b_{n,r-2})\quad (r>2) $$ ...

Where does the following formula come from? For a Laurent polynomial $f(z)=\sum a_j z^j$ and a positive integer $n$ we have $$\sum_{k\equiv \alpha\pmod n} a_k=\frac1n\sum_{\omega:\omega^n=1} \omega^{-...

Pick four integers $a,b,c$ and $d$. Then we get a corresponding sequence given by $$t_{n+2} = at_{n+1} +bt_n, \; t_1 = c, \;t_2 = d.$$ From what I can tell, we seem to get an especially rich theory ...

The exercise: A group of N friends sits around a table shaped as a regular polygon with N sides, one person on each side. Everyone tosses a fair coin once and a person is called positive if she and ...

Prove: the number of simple undirected graphs with n different nodes, s.t every node has even degree is $2^\binom{n-1}{2}$ My attempt: I think the problem is equivalent to the number of $(n \times n)...

Just as the question asks. I am trying to calculate the number of bit strings of length $n$ with a maximum of $k_1$ consecutive $0s$ and $k_2$ consecutive 1s. Of course we assume $k_1+k_2\leq n$. I am ...

Consider an urn which contains three different kinds of balls A, B and C. We suppose that there is at least one ball of each kind in the urn. We define the event $A$ as "to get, in $n$ independent ...

We arrange $9$ balls numbered $1,\dots,9$ in a row randomly (a permutation). Let $X_i$ an indicator to the ball in the $i$ position is less than the ball in the $i+1$. Prove $\mathbb{E}[X_2|X_1]=\...

How many triangles can be formed by the vertices of a regular polygon of $n$ sides? And how many if no side of the polygon is to be a side of any triangle ? I have no idea where I should start to ...

Given a sequence $1,1,2,2,3,3, …,k,k$, I am interested in counting the number of non-nesting permutations of the above sequence. Two intervals (determined by symbols $K$ and $L$) are nesting if one is ...

The following notations and definitions are taken from Richard Stanley's book Enumerative Combinatorics Volume $1,$ second edition. Recall that a formal power series $F(x)$ is of the form $$\sum_{n\...

How to algebraically prove that $$n!=\sum\limits_{r=0}^n (-1)^r \binom{n}{r} (n-r)^n$$ I was trying to find number of onto functions from $A$ to $A$ containing $n$ elements. Using the inclusion-...

Let $X=[X_1, \dots, X_n]$ be a random vector of binary-valued random variables, taking values in $\{0,1\}^n$. Let $S(n,k)$ denote the set of all size $k$ subsets of $\{1,\dots,n\}$. Can anyone ...

To find the expected number of suits the formula is $E(Num Suits) = 1*P(1 Suit) + 2*P(2 Suit) + 3*P(3 Suit) + 4*P(4 Suit)$ For the probability of getting 4 suits I got ${13 \choose 1}^4 {4 \choose 4}/...

Consider three events $A,B,C$ such that $P(A)>0$, $P(B)>0$, and $P(C)>0$. The events are linked to each other through the constraints $P(A\cup B\cup C)=1$ and $P(A)=P(\overline{B})$. Under ...

Given a multinomial distribution with parameters $n>0$ where $n$ is an integer and event probabilities $p_i= 1/k$ for $i \in \left\{1, \ldots, k\right\}$. Next, allow that $\mathbf{N}$ is a ...

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