# 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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### Sum of digits of permutations and combinations of a given set of digits [closed]

What is the sum of all $5$-digit numbers formed from $\{2,3,4,4,6,0\}$ without allowing repetition? What is the sum with repetition allowed?
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### Partition lattice-maximal chains

Show that the number of maximal chains in the partition lattice $\prod _n$ is equal to $\dfrac{(n-1)!n!}{2^{n-1}}$. I showed that $\prod _n$ is graded lattice, so all maximal chains has the same ...
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### How to calculate sum of LCMs [duplicate]

How to solve this problem? Given n, calculate the sum LCM(1,n) + LCM(2,n) + .. + LCM(n,n). Is there any way to solve it by math?
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### Find all Functions so that $f(1) = 1$ and $f(2) = 2$

Let $F$ denote the set of all functions from $A=\{1, 2, 3, 4\}$ to $B=\{1, 2, 3, ..., 10\}$. Find and simplify the number of functions $f \in F$ so that $f(1) = 1$ and $f(2) = 2.$ My attempt to ...
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### 2014 iberoamerican olympiad Problem 3

2014 points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of ...
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### Probability of Group Standings vs. Randomized Standings

This question concerns MLB baseball standings. There are 6 divisions with 5 teams each for 30 total teams. Currently one division has three of the four best records. What are the odds that this ...
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### Find the number of all isosceles triangles, where all three vertices belong to the set $\{A_1,A_2, \cdots,A_{30}\}$

In the coordinate plane let $A_i=(i,1)$ for $l\leq i\leq15$, and let $A_i=(i-15,4)$ for $16\leq i \leq 30$. Find the number of all isosceles triangles, where all three vertices belong to the set ...
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### The probability that each delegate sits next to at least one delegate from another country

Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate ...
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### Number of $k$ subsets of $S$ by choosing $i$ elements from $A$ and $j$ elements from $B$ where $S=A \cup B$

Let $A$ be a set with $m$ elements and let $B$ be a set with $n$ elements. Let $S=A \cup B$. Then the number of $k$-subsets of $S$ is clearly $C((m+n),k)$. However, if we want the number of $k$ ...
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### Proving binomial coefficient formula based on Pascal's triangle

I am trying to practice proving things, and I came across one I wasn't sure about. We already know that $\binom{n}{k}$ is the sum of the two corresponding "parent" entities in Pascal's triangle, ...
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### Find a recurrence to count paths in a directed graph

Suppose we have an unweighted directed graph with vertices numbered as $1...n$ From each vertex $i$ there are edges to $i+1$, $i+2$ and $i+7$. My task is to find a recurrence $f(i,j)$ to compute the ...
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### Find the total number of selections of r things from n different things when each thing can be repeated unlimited number of times?

Find the total number of selections of r things from n different things when each thing can be repeated unlimited number of times ? I know that the formula is $$n+r-1\choose r$$ But how do we get ...
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### Number of Non - Decreasing functions?

Let A={1,2,3.....10} & B={1,2,3....20}. We have to find the number of non decreasing functions from A-->B. What I tried :No. Of non decreasing functions = (Total functions) - (Number of ...
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### How to solve/approach for counting the large range of numbers in mind for this particular type of eliminating numbers?

Here is the following question. I was wondering on how to solve such questions. 100 people standing in a circle in an order 1 to 100. No 1 has a sword. He kills next person i.e. No 2 and gives sword ...
### Probability of hitting a number - $\mathsf{II}$
Suppose you have $\frac{cn}{(\log c+\log n)^a}$ distinct pairs of numbers where fixed $c,a$ satisfies $1<c,a<\infty$. You are to choose two sets of $\frac{4\sqrt{n}}{(\log n)^b}$ distinct pairs ...