# Tagged Questions

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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### Need to create a list of combinations from a given set

I need the best way to create a list of all the combinations for a given set so each element in the list sits next to all others with the least number of repeated combinations. Example: I have x ...
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### Probability of alternating shirts

5 men are wearing red shirts and 5 men are wearing blue shirts. If the 10 men are lined up randomly, what is the probability that the colors will alternate? My attempt: I started out by assuming ...
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### Ramsey number for tree and complete graph [duplicate]

I am having a lot of trouble understanding Ramsey theory. I am working on an exercise that asks for the Ramsey number $R(T,K_{1,n+1})$ where $T$ is a tree with $m$ edges and $n$ is a multiple of $m$. ...
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### Combinatorics: Using binomial coefficients to figure out playing card combinations

I have this sample problem from my notes about how to find different 5-card poker hands from a standard 52-card deck. Can someone explain to me what's going on? I don't get why the ...
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### Multiple persons flip biased coins multiple times

I've been trying to solve this seemingly easy problem for some time, but I'm not that well-versed in probability, so I thought I would ask. Let there be $N$ persons each with identical biased coins ...
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### A problem of periodic functions , the greatest common divisor and a lattice

I am trying to solve the following problem. If $\psi(s) = \frac{s(s-1)}2$. I write $f(s,k) = (\psi(s),\psi(s-k))$, where $k$ is a fixed positive integer. Let $K$ be the image of $f_{s,k}$. If $s>3$...
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### The Hand Shaking Lemma

In any graph G=(V,E) [the hand shaking lemma] $$\sum_{v \in V} \deg(v) = 2 |E|$$ (original at http://i.stack.imgur.com/af4en.png) where |E| donetes the number of edges I alredy tried to count ...
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### Number of object distribution between four people (with cases)

Four people are dealing the total amount of money, which is $1000$ monetary units in terms of $100$ monetary units. Count the number of ways for this distribution if: $1)$ Every person doesn't have ...
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### How many of the spanning trees, $K_n$ have vertex n as a leaf?

So I know that I should probably use Cayley's formula here, which is that for positive integer $n$, there are $n^{n-2}$ labeled trees on n vertices. So I looked at a few trees and saw that when n = 3 ...
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### Combinatorics How Many Tree

How can I prove that for any tree $G=(V,E)$, $$|E|=|V|-1$$ I have tried the induction on the number of vertices but nothing happened.
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### Sum of the first k multi-nomial coefficients for fixed n

Multinomial coefficients are defined as $$\binom{n}{k_1,k_2,\cdots,k_m}$$ where $n=k_1+k_2+\cdots+k_m$ Is there closed form solution available for the sum of the first $t$ multinoial coefficients? ...
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### Combinatorics - # of ways to choose people to 2 groups with condition

In a class there are 30 students, we need to choose 2 groups of 11 students so they can play against each other. Josh, one of the pupils has to be in one of the groups. What I did is this: We'll put ...
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### Choose Sixteen Cookies from Five Varieties

A cookie shop sells 5 different kinds of cookies. How many different ways are there to choose 16 cookies if... (a) you have no restrictions? (b) you pick at least two of each? (c) you pick at ...
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### How should a composite variable be constructed?

I have a set of multiple variables which are used as the arguments of a function. I have collected all of the instances of the various values of the variables together with the output of the function ...
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### Existence of a (40,13,4)BIBD (Balanced Incomplete Block Design)

I have been asked to prove that there exists a (40,13,4)BIBD. I admittedly have no idea where to start with this. Checking some of the necessary conditions for BIBDs shows me that if such a BIBD ...