Questions tagged [combinatorics]
For questions about the study of finite or countable discrete structures, especially 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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How to find all possible solution sets to this simple addition problem?
Say I have $5$ non-negative integers $a,b,c,d,e$.
I need $a+b+c+d+e=s$. I'd like to find a way get all possible sets of answers. E.g. $[10,0,0,0,0], [9,1,0,0,0]$, etc. I was thinking of brute forcing ...
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Selecting three non consecutive vertices from a regular decagon
A regular polygon of $10$ sides is constructed. In how many ways can 3 vertices be selected so that no two vertices are consecutive.
Attempt:
Choose 1 vertex -> mark the two vertices next to it as ...
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Number of combinations of ice-cream with limited quantity.
I am trying to solve the following problem on combinations:
Ten people visit an ice-cream shop that sells the following ice-cream
flavours: Apricot, Banana, Cherry, and Apple.
There are ...
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Find the number of 3-letter words that can be made with letters in alphabetical order.
Consider the first ten letters of the alphabet $\{A,B,C...J\}$ and consider any three letter sequence a word. How many three letter 'words' can be constructed from this set in which all the letters ...
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Use inclusion/exclusion to find the number of derangements of each of the following strings. [duplicate]
a) aabcd (already answered)
b) aabbcc
There is a problem exactly like the one I asked, however I am still stuck and still need help. Here is my attempt
Let A denotes where aa occurs, B denotes ...
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probability of drawing a 5 card flush given n cards [closed]
Given $n$ random cards from a standard $52$ card deck, what is the probability of getting at least a 5 card flush within those $n$ cards? $n$ would be 5 <= $n$ < 17. The probability would get ...
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Permutation of numbers in a specific order.
How to calculate number of permutations from $1$ to $N$ such that for some $j$ these properties hold :
$$
P(i)>P(i-1)\;\;\; \
$$
$$
P(i) > P(i+1)\;\;\; \
$$
For $N=3$ it can be $(1,3,2)$
For $N=...
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Combination on the different sum of money [duplicate]
How many different sums of money can be made from penny, a nickel (5 pennies), a dime (10 pennies) and a quarter (25 pennies)?
I am having trouble sorting out the answer. As far as I have made ...
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Help inequality real numbers
I'm currently reading the book "The Design of Approximation Algorithms". In page 25, they introduce Fact 1.10 and use this to prove Theorem 1.1. Below stated is Fact 1.10.
Fact 1.10 Given positive ...
user394255
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Please vet analysis for wrong way of computing choices for three-of-a-kind.
It is given on page #21 of U.G. book : Combinatorics - A guided Tour, by David Mazur, here an example detailing chances of getting three-of-a-kind in 5 card hands.
The problem statement is :
From a ...
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Show that cardinality of intersection is infinite
For each positive real number $x$ let $S(x)=\{\lfloor kx\rfloor \,:\, k \in\mathbb{N}\}$. Let $x_{1},x_{2},x_{3}$ be positive real numbers each greater than $1$ such that $\sum_{i=1}^{3}\frac{1}{x_i} &...
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Question on cross-cuts contained in cross-cuts
Let $X$ be a finite set with $|X| =n $ and $\mathcal{A} \subset \mathcal{P}(X)$ a set system.
Call $\mathcal{A}$ a cross-cut if $\forall B \in \mathcal{P}(X), \; \exists A \in \mathcal{A} $ such ...
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Determine the Number of Multiples of Given Numbers $\le$ 1000
For one of the problems I am currently trying to solve, I am to determine the number of multiples of 3, 5 and 7 that are between 0 and 1000. Also, I am to do the same for each combination of the three ...
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Let $n$ be a positive integer, Prove that $\sum_{k=1}^n\frac{ (-1)^{k-1}}{k}{n \choose k} = H_n$
Let $n$ be a positive integer, Prove that $\sum_{k=1}^n \frac{(-1)^{(k-1)}}{ k} {n \choose k} = H_n = 1+\frac{1}{2}+\ldots+\frac{1}{n}.$
This problem was solved as an example in Titu Andreescu's ...
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How many numbers in $\{2,3,...,360\}$ share at least one prime factor with $360$?
What is the best way to go about solving this question?
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