# 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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### Using Routes To Map Increasing Mappings

Problem So how do I establish a bijection between these two sets? Also, $N_n$ = (1,2,3,4,...,n). Thank you.
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### Coloring the pentagonal hexecontahedron

So, I'd like to color the pentagonal hexecontahedron in a way that is satisfying aesthetically and mathematically. For me this equates to, in order of priority - 1. No same-colored faces can share an ...
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### What is the probability of a random 8 bit string to have no more than 2 consecutive 1's. [closed]

I don't know how to approach this problem. I think the correct approach is getting a recurrence relation. But I don't know how. Help is much appreciated. This is not a homework problem. I saw ...
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### Checking if something is a Bijection

Reflection Principle's Proof I was able to follow the proof until the end, and then the proof said to check that it was a bijection. How would one check if something was a bijection?
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### How many combinations to break a monoalphabetic substitution

Let a language $\Sigma$ have 16 letters, we have a message in that language that was encrypted using monoalphabetic substitution (a permutation of the alphabet) and we want to break it. We also ...
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### Increasing Mapping [duplicate]

Problem What does it mean when by a strictly increasing mapping? For example, if you had $8$ = (1,2,3,4,5,6,7,8) and $3$ = (1,2,3) what would the increased mapping be?
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### Solving $x_1 + \dots + x_n = m$ with general (i.e. not specific to a variable) restrictions

The number of non-negative integer solutions to $$x_1 + \dots + x_n = m$$ is extremely well known to be ${m + n - 1 \choose m}$. It is also not difficult to solve if we require, say, $x_1 \geq 5$: ...
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### 8 people in 4 teams with different pairs in each team each day for 7 days without repeated pairs or anyone being in the same within 3 days

Ok I am a Scout Leader and on our 7 day summer camp we have 8 Leaders and will have the Scouts in 4 different patrols or teams. I want to set up a rota for the Leaders so that they can be assigned to ...
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### Number of ways to arrange $n$ numbers based on their relative values to each other

EDIT I've found a formula to solve this question, but I don't understand the reasoning behind it. Can someone explain this formula? $s(n - 1, x + y - 2) \times C(x + y - 2, x - 1)$ $s$ being ...
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### How many line segments have both their end points located at vertices of a given cube?

How many line segments have both their end points located at vertices of a given cube? My try:- A cube has 8 vertices. Number of line segments = 8C2=28. (As a line segments has 2 end points)
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### Find permutation matrix $X \in \{0,1\}^{N \times N}$ in order to make $XAX \geq_c B$

I need to solve a problem to find out the best permutation matrix $X \in \{0,1\}^{N \times N}$ which would maximize the number of elements in matrix $XAX$ which are above (component-wise) matrix $B$ ...
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### A Closed Form Lower Bound Approximating $p_{n,m,s} = n![z^n]\left(\sum_{k=0}^s\frac{z^k}{k!}\right)^m$

Here, I found $p_{n,m,s} = n![z^n]\left(\sum_{k=0}^s\frac{z^k}{k!}\right)^m = \sum\limits_{\substack{k_1 + \cdots + k_m=n\\0\leq k_i \leq s}} \frac{n!}{k_1!\cdots k_m!}$ as the number of ways to ...
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### Compute the number of ordered partitions [closed]

Let $a-b=2n$. Compute the number of ordered partitions of an integer $a$ if they include $b$ odd numbers.
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### Prove there's a monochromatic isosceles triangle.

The points in a circle are coloured red and blue. Prove that there exists a monochromatic isoceles triangle. I can prove that there exists a monochromatic triangle. If there are no three points of ...
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### Coefficient of $x^n$ in $x \prod\nolimits_{i = 1}^{d} (x - i) (x + i)$

I am looking for a general form of the coefficient of $x^n$ in $x \prod\nolimits_{i = 1}^{d} (x - i) (x + i)$. I know that the leading coefficient (in front of $x^{2 d + 1}$) is $1$, and there are no ...
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### Condition for n points in the plane to determine a convex n gon

Suppose there are n points in the plane, labelled 1 through n, no three of which lie on a line. Suppose further that for every triple [i,j,k] with i< j < k that travelling from i to j to k is ...
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### $k$ kids, father and mother, it is forbidden to sit two chosen kids near to each other, How many ways there are to arrange them in line?

I have the following question : We have $k$ kids, father and mother, it is forbidden to sit two chosen kids near to each other. How many ways there are to arrange them in line? This is what I ...
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### How to solve this Iran TST 2014,second exam, problem？

This Problem is Iran TST 2014, second exam, day 2 ,problem 3 Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following ...
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