# 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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### Plaid in generic position. Counting faces.

I write $\pi_n$ to denote a group of $n$ parallel lines. Consider a family of $(\pi_1,\pi_2,\ldots,\pi_s)$ parallel groups each with $(n_1,n_2,\ldots,n_s)$ parallel lines. Arrange the family of ...
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### Method of integration [duplicate]

We have to find the integration of the following function I tried but got stuck can anybody help me how to proceed . Is there anyother method to solve this
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### Optimization Algorithm for Combining Nodes on a Graph

Graph before and after clustering nodes $R_1$ and $R_2$ In the picture linked above, I have a graph with nodes $R_1$ through $R_5$ and vertices linking them. All the vertices are weighted 10 in this ...
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### Counting solutions by estimating Fourier coefficients

In W. T. Gower's essay The Two Cultures of Mathematics, he mentions the following as an example of a 'general principle' in combinatorics: "If one is counting solutions, inside a given set, to a ...
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### How many distinct ways are there to $2$-color the $8$ vertices of a cube?

How many distinct ways are there to $2$-color the $8$ vertices of a cube, with colorings only considered distinct up to rotation?
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### lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set$S = \{ |s(\vec x)| : \vec x \...
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### Closed form of recurrence relation $F(n) = 2 + F(n-1) + F(n-2)$

I was figuring out an answer to the question, How many Boolean arrays of length $n$ could be formed if there are to be no two falses in a row? I could see that it boils down to a Fibonacci ...
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### Applications of tensor product of graphs (modelling of Internet Graphs)

I was going through the book Handbook of Product Graphs, by Richard Hammack, Wilfried Imrich, Sandi Klavžar. Somewhere in book, they mentioned the following lines : One of the applications of tensor ...
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### find the number of permutations of the letters of the word ANTENNA taken 4 at a time? [closed]

I understand how to get the permutation of the word itself, but what does the "taken 4 at a time" mean?
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### determine the number of poker hands that are better than 2 Aces, 2 Eights, and a 5?

I am not very familiar with the game of poker so I have no clue where to begin answering this question. it is along the lines of using combinations to solve though.
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### excecutives from 25 student clubs, one male and one female…

excecutives from 25 student clubs, one male and one female from each are attending a workshop on student violence. how many ways can a commitee be set up of 5 men and 7 women if only one male or ...
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### Probability with biased coin problem

Jules César gives Astérix a biased coin which produces heads 70% of the time, and asks him to play one of the following games: Game A : Toss the biased coin 99 times. If there are more than 49 heads, ...
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### In how many ways can 10 different things be distributed to 4 persons if 2 are to receive 2 things and the others are to receive 3 things?

I have no idea how to answer this question, I did a lot of research on trying to figure it out but every answer is so different. I would prefer something along the lines of using combinations and ...
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### Counting number of bases in a set of vectors with spanning constraint

I am interested in a good bound for the following problem: Suppose $S = \{ v_1, \dots, v_n \}$ is a set of vectors where $v_i \in \mathbb{R}^r$ for $i = 1, \dots, n$. Suppose further that any ...
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### What is the rank of COCHIN

Is there any shortcut method for finding the rank of the word COCHIN? I mean is there any shortcut method for finding the rank of a word having repeated letters. For example there is a shortcut method ...
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### every tree $T$ has at most one perfect matching, alternative proof

I have two questions: I need to know if the following approach (by induction) is correct. The ones I saw use induction on the components of $T$ with a leaf removed, I did something a little different....
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