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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 \(f(x) = (1-4x)^{-1/2}\).
Now you Nijedan kalkulator ne slaže ovaj, ali komadiće su komputentni. Pokušajte jedan ispod, ili upišite vlastiti.
Besplatan račun dodaje bilješke na svakoj lekciji, zapis onoga što ste završili, rješeni problemi na jednom mjestu, i tutor možete pitati o ovoj stranici. Sama matematika je otvorena za sve, potpisani ili ne.
Prijavi se PrijavaSimboli koji se koriste ovdje
Dodirnite bilo koji simbol za punu definiciju, sliku, i što znači svako slovo u njoj.
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.
Dijelovi ove stranice prilagođeni su od Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Ovdje su kondenzirane i objašnjene; 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