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Combinatorics and Geometry
There are many problems in geometry that are innately combinatorial or for which combinatorial techniques shed light on the problem. ExampleIn , we show a family of 4 lines in the plane.
Combinatorics and Geometry
There are many problems in geometry that are innately combinatorial or for which combinatorial techniques shed light on the problem.
Example
In , we show a family of \(4\) lines in the plane. Each pair of lines intersects and no point in the plane belongs to more than two lines. These lines determine\(11\) regions.
Under these same restrictions, how many regions would a family of \(8947\) lines determine? Can different arrangements of lines determine different numbers of regions?
Example
Mandy says she has found a set of\(882\) points in the plane that determine exactly \(752\) lines. Tobias disputes her claim. Who is right?
Example
There are many different ways to draw a graph in the plane. Some drawings may have crossing edges while others don't. But sometimes, crossing edges must appear in any drawing. Consider the graph \(G\)
Can you redraw \(G\) without crossing edges?
Suppose Sam and Deborah were given a homework problem asking whether a particular graph on \(2843952\) vertices and \(9748032\) edges could be drawn without edge crossings. Deborah just looked at the number of vertices and the number of edges and said that the answer is no. Sam questions how she can be so certainwithout looking more closely at the structure of the graph. Is there a way for Deborah to justify her definitive response?
Sada ti Nema kalkulatora koji bi riješio ovaj problem, ali njegovi dijelovi se mogu izračunati. Pokušajte jedan od gore navedenih ili upišite svoj.
Besplatni račun dodaje bilješke na svaku lekciju, zapis onoga što ste završili, vaše riješene probleme na jednom mjestu, i tutora kojeg možete pitati o ovoj stranici. Sama matematika je otvorena za svakoga, prijavljenog ili ne.
Prijavite se PrijavaSimboli korišteni ovdje
Dodirnite bilo koji simbol za potpunu definiciju, sliku i što svako slovo u njemu znači.
Pitanja koja ljudi postavljaju
Permutation or combination?
Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).
What is a graph in this sense?
Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.
Dio ove stranice je prilagođen sa Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). U ovom slučaju, greške su naše.
Više u Combinatorics & Graph Theory
The counting principlesPigeonhole principle and inclusion, exclusionBinomial coefficients and Pascal's triangleRecurrences and generating functionsGraphs: vertices, edges, degreesPaths, cycles, trees, Euler and HamiltonColouring and planar graphs