Articles of combinatorics

pigeonhole principle related problem

I’m given the problem: In a tournament which 18 teams participate, a team being matched with another in a round don’t match again in the follwoing (later) rounds. After 8 rounds prove that there are 3 teams not being matched with each other. I don’t know where to start from. Can anyone help me to […]

Ice cream combinatorics question

An ice cream shop sells ice creams in $5$ possible flavours: vanilla, chocolate, strawberry, mango and pineapple. How many combinations of $3$ scoops cone are possible? [note: repetition of flavours is allowed, but the order in which they are chosen does not matter.]

Proof on distinct odd parts, partitions

Let $p (n|$distinct odd parts$)$ be the number of partitions of $n$ into distinct odd parts. Prove that $p(n)$ is odd if and only if $p(n|$distinct odd parts) is odd. I know we’re suppose to use the theorem on self-conjugate partitions…

Chess combination

On the chess board 8×8, we have 8 Rooks that , each Rook don’t atack the other. How many situation if we give out: a) 1 main diagonal b) 2 main diagonals That’s mean we can’t put the Rook to the cells of diagonal

A classroom has two rows of eight seats…

A classroom has two rows of eight seats each. There are 14 students, 5 of who always sit in the front row and 4 of who always sit in the back row. In how many ways can the students be seated? $$\dfrac{8!}{3!}\dfrac{8!}{4!}\dfrac{7!}{2!} = 28449792000$$ The correct answer might be clearly obvious but I’m asking what […]

Upper bound for ramsey number $r(a_1,\ldots, a_m)$

I am looking for any (finite) upper bound of the ramsey number $r(a_1,\ldots, a_m)$. I can prove the well known fact for any positive integers $a,b$ there is a $c$ for which $c\ge r(a,b)$ by taking $c=r(a-1,b)+r(a,b-1)$. Is this argument and its proof restricted to 2-tuples or would $r(a_1-1,a_2,\ldots ,a_m)+\ldots +r(a_1,\ldots,a_m-1)$ be an upper bound […]

Number of possible routes through n countries and 2n cities, with restrictions

Someone is planning a round-the-world trip that involves visiting $2n$ cities, with two cities from each of $n$ different countries. He can choose a city to start and end the journey in, with the other $2n-1$ cities being visited exactly once. However, he has the restriction that the two cities from each country should not […]

I've got the same answer to my question via probability and combinations, but I don't know why!

I am trying to find the answer to the following question: If I am playing poker and am dealt two cards neither of which are spades, what are the chances of two or more of the flop cards (the flop is the first three community cards) being spades. I answered this question in the following […]

Binary Strings of the form *111*

Possible Duplicate: Counting subsets containing three consecutive elements (previously Summation over large values of nCr) Given an integer $N$, we have to count the number of possible binary strings(string is madeup of only $0$’s and $1$’s) of length $N$, and matches the regular expression pattern $[111]$. For example if $N$ is $4$, then $1110$ , […]

Scheduling a team based activity with 10 different teams and X different games. Every team must meet each other ONCE but never twice,

Hi there I am planning a ”Scavenger hunt”-like activity (I have no other words for this event), and I am having trouble creating the schedule. The idea of this hunt/run/game/competition is that 10 teams will battle each other at X different mini games. The games on are team vs. team based, meaning that there will […]