# 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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### number of lattice points in an n-ball

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems. Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that ...
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### Counting Lattice Points with Ehrhart Polynomials

Let $\bar{\mathcal{P}}$ denote the closed, convex polytope with vertices at the origin and the positive rational points $(b_{1}, \dots, 0), \dots, (0, \dots, b_{n})$. Define the Ehrhart quasi-...
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### Counting lattice points interior to a polygon

If I define an integer lattice $\Lambda \subseteq \mathbb{Z}^2$ with a basis given by $$\omega_{1} = a \hat{i} + b\hat{j}, \;\;\; \omega_{2} = -b \hat{i} + a\hat{j}$$ How can I count how many lattice ...
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### Find least number of radial-subgraph of a graph

Background: Here is a group G of a people, one maybe another's friend. How to select least number of people to be a leader of a subgroup, so that everyone in the group G has a friend as a leader? ...
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### In AB + BC + AC = N, how can I find all possibilities for A, B and C in less than n³ computational time?

The problem is the one on the title. Given a N, find all possibilies for A, B and C that make this true: $AB+BC+AC = N$when $A \ge B \ge C$. This code in C do the job: ...
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### Bayesian probability problem?

Problem: In a city there are three types of taxis which drive towards the airport. 30% are blue, 20% green, 50% yellow. They take there customers too late with probabilities 0.1,0.2,0.3 respectively. ...
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### Prove that if a collection of subsets of {1,..,n} that each pair of subsets has at least one element in common, there are at most $2^{n-1}$ subsets

Full question: Prove that if a collection of subsets of {1,2,...,n} has the property that each pair of subsets has at least one element in common, then there are at most $2^{n-1}$ subsets in the ...
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### Combinatorial Identity with Binomial Coefficients: ${{a+b+c-1}\choose c} = \sum_{i+j=c} {{a+i-1}\choose i}{{b+j-1}\choose j}$

I got the following identity from commutative algebra. I am curious to see elegant elementary methods. $${{a+b+c-1}\choose c} = \sum_{i+j=c} {{a+i-1}\choose i}{{b+j-1}\choose j}$$
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### Find a binomial coefficient equal to ${n\choose k} + 3 {n\choose k-1} + 3{n \choose k-2} + {n\choose k-3}$

Exercise. Find a binomial coefficient equal to: $${n\choose k} + 3 {n\choose k-1} + 3{n \choose k-2} + {n\choose k-3}.$$ I don't really understand what we are asked to do when we are told to ...
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### How to evaluate the sum of binomial coefficients $\sum_{k=m}^n {k\choose m}^2$?

I know the following identity according to wiki (the one before eq (9)) $$\sum_{k=m}^n {k\choose m}={n+1\choose m+1}$$ Is there an identity for the following sum? $$\sum_{k=m}^n {k\choose m}^2$$ ...
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