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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 \(f(x) = (1-4x)^{-1/2}\).

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Àwọn Àtòjọ-ẹ̀yàn

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.

Àwọn ààyè ojú-ìwé yìí tí a fi pamọ́ láti Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Tí a fi pamọ́ síì níbẹ̀; àwọn àwọn àṣiṣe ní wa.

Diẹ̀ nínú Combinatorics & Graph Theory

The counting principles Pigeonhole principle and inclusion, exclusion Binomial coefficients and Pascal's triangle Recurrences and generating functions Graphs: vertices, edges, degrees Paths, cycles, trees, Euler and Hamilton Colouring and planar graphs