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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Pigeonhole Principle Question: Jessica the Combinatorics Student

Jessica is studying combinatorics during a $7$-week period. She will study a positive integer number of hours every day during the $7$ weeks (so, for example, she won't study for $0$ or $1.5$ ...
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How many integers in $\{500,…,1000\}$ are not divisible by 3, 7 or 13?

I am wondering what the best way to approach this question is. I thought that I would calculate the number of integers that aren't divisible by 3, 7 or 13 in $\{1,2,...,1000\}$ as well as the number ...
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Coloring a triangular bipyramid

A triangular bipyramid looks like this: http://mathworld.wolfram.com/TriangularDipyramid.html I have to find the ways to color it using n colors allowing rotations and reflections. I do not ...
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Bridges across a tiled floor

A few years back, a friend of mine did a seminar on "Bridges across a tiled floor". A "bridge" was defined as a row or column of an $n \times n$ binary matrix consisting entirely of $1$'s, for ...
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In how many ways can 8 similar rings be worn in five fingers of a hand? [closed]

Provided that a finger may not contain more than one ring.However a finger may be empty.
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Let $S$ be the set numbers whose digits are chosen from ${1, 3, 5, 7}$ such that no digits are repeated. Find the sum of every element in $S$.

All numbers in $S$ are natural. I could find the $|S| = 64$ by my own. Can't find the sum of every number in $S$, nor understand the book's explanation for that. The answer is $117856$. Taken from ...
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Summation with combinations

Prove that $n$ divides $$\sum_{d \mid \gcd(n,k)} \mu(d) \binom{n/d}{k/d}$$ for every natural number $n$ and for every $k$ where $1 \leq k \leq n.$ Note: $\mu(n)$ denotes the Möbius function. I have ...
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Supposedly really hard problem involving combinations

This problem gives 7 (max) out of 100 points for a college entrance exams. Seems odd because it looks easy to me, although my combinations are not too good. There are $10$ people forming a ...
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Degree of Jacobian of homogeneous polynomials

What is the degree of the Jacobian (as a polynomial) of 3 homogeneous polynomials in 3 variables of degrees say $m_1, m_2$ and $m_3$ ? I don't know how to prove that it is independent. In my case the ...
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Number of even numbers having digit 2 in them.

I am trying to count numbers from 1 to N which exist in A121022 but I am unable to think of solving in better than O(NLog(N)) , can you suggest a better algorithm?
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Probably an ambiguous word problem

I don't know if this should have been posted on English because it's about interpretation of a sentence, or Math because it involves with a math problem to get the right context and interpretation... ...
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Combinations of sandwiches

My stats summer packet proposes the question: "if a sandwich shop has $3$ different types of meat, $4$ different types of bread, and $3$ different types of cheese. How many types of sandwiches can you ...
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mutual information and combinatorics

\begin{align} &\mathrm{H}\left(\frac{1}{2^{k}}\right) \\[3mm]&\ \!\!\!\!\!\!\!\!\!\! - {1 \over 2^{k}}\left\{% {k \choose 0}\mathrm{H}\left(\left[1 - \epsilon\right]^{\,k}\right) + {k \choose ...
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Is this equation true?

As the question states, does this equation hold true? $\sum_{j=0}^n \sum_{E \in {n \choose j}} (-1)^{|E|}(n-|E|)! = \sum_{j=0}^n(-1)^j(n-j)!{n \choose j}$ From what I understand, this holds true at ...
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Numbers with constant digit-sum in increasing order

For base $b = 10$, I want to list all numbers with $d$ digits (no leading zeros) and digit sum $x$, in increasing order. For example for $d = 6$ and $x = 40$ we would get: 139999, 148999, 149899, ...
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Probability of 4 specific numbers (1-3000) occuring in a sample of 400

How to calculate the probability that four specific, distinct numbers from the range 1 - 3000 occur at least once in a fixed sample of 400 random numbers from the range 1-3000? The numbers in the ...
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Existence of Kazhdan Lusztig basis proof due to Soergel

This question is regarding the proof of the existence and uniqueness of Kazhdan Lusztig basis theorem for an arbitrary coxeter group $W$ due to Kazhdan and Lusztig in his paper "Representations of ...
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Number of licence plates that match a criterion [closed]

A new license plate in Alberta consists of three letters followed by four numbers. Letters are chosen from a list of $24$ acceptable letters that may be repeated. And any digits can be used and they ...
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When arranging numbers and letters in combinatorics, should one use multiplication or addition?

Let's say that we are given that a code is formed with 3 letters of alphabet followed by 3 digits from 0-9, and both can be repeated. When required to find the total number of combinations. Is it ...
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stirling numbers of second kind

i am new to combinatorics and just encountered stirling numbers of second kind the book i am using does not provide much info about it except number of ways of distributing "r" distinct objects ...