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The Binomial Theorem

Here is a truly basic result from combinatorics kindergarten. Theorem: Binomial TheoremLet x and y be real numbers with x, y and x+y non-zero.

The Binomial Theorem

Here is a truly basic result from combinatorics kindergarten.

Example

There are times when we are interested not in the full expansion of a power of a binomial, but just the coefficient on one of the terms. The gives that the coefficient of \(x^5y^8\) in \((2x-3y)^{13}\) is \(\binom{13}{5}2^{5}(-3)^8\).

Nígbà yìí Kò ní kọ́ǹpútà alágbèéká kan tí o lè fi àwọn à

Àwọn Ìṣàmúlò-ètò (1)

Wá gbogbo àwọn ní pàtó nínú àwọn àkọ́lé ní ìbẹrẹ. Fi àwọn

Explain how you can use to give yet another proof of the identity \[\begin{aligned}\end{aligned}\].

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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