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An Application of the Binomial Theorem
In this section, we see how can be used to derive another useful identity. We begin by establishing a different recursive formula for P(p,k) than was used in our definition of it.
An Application of the Binomial Theorem
In this section, we see how can be used to derive another useful identity. We begin by establishing a different recursive formula for \(P(p,k)\) than was used in our definition of it.
Our goal in this section will be to invoke with the exponent \(p=-1/2\). To do so in a meaningful manner, we need a simplified expression for \(C(-1/2,k)\), which the next lemma provides.
We will return to this generating function in , where it will play a role in a seemingly new counting problem that actually is a problem we've already studied in disguise.
Now recalling about the coefficients in the product of two generating functions, we are able to deduce the following corollary of by squaring the function \(f(x) = (1-4x)^{-1/2}\).
Nüüd sina. Ükski kalkulaator ei lahenda seda, kuid selle osad on kompueeritavad. Proovi allpool või kirjuta omaenda oma.
Tasuta konto lisab märkmeid igale õppetunnile, ülestähenduse, oma lahendatud probleemide kohta ühes kohas ja juhendaja kohta, kellelt võid küsida selle lehekülje kohta. Matemaatika ise on avatud kõigile, allkirjastatud või mitte.
Registreeru SisselogimineSiin kasutatavad sümbolid
Koputage kõiki sümboleid, mis on mõeldud täieliku definitsiooni, pildi ja iga tähe all.
Küsimusi, mida inimesed küsivad
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.
Osad käesoleva lehekülje on kohandatud alates Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Kondenseeritud ja uuesti selgitatud siin; vead on meie.
Rohkem 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