The Binomial Theorem

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 \(\binom{13}{5}2^{5}(-3)^8\).

Issa inti L-ebda kalkulatur ma jista' jsolvi din il-problema, iżda l-biċċiet tagħha jistgħu jiġu kkalkulati. Ipprova waħda minn hawn taħt, jew ittajpja tiegħek stess.

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Problema matematika tiegħek

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Prattika (1)

Ipprova kull waħda fuq il-karta l-ewwel.Iżvela t-tweġiba biex tiċċekkja; dawk ivverifikati jistgħu jinfetħu fis-solver għal kull pass.

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

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Żomm tiegħek stess li jaħdmu

Kont b'xejn iżid noti fuq kull lezzjoni, rekord ta' dak li lest, problemi solvuti tiegħek f'post wieħed, u għalliem inti tista' tistaqsi dwar din il-paġna.Il-matematika nnifisha hija miftuħa għal kulħadd, log fil jew le.

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Simboli użati hawnhekk

Taptap kwalunkwe simbolu għad-definizzjoni sħiħa, stampa, u dak li kull ittra fiha tfisser.

\binom{n}{k}binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).iimaginary unit
i² = −1.n!factorial
n × (n−1) × … × 1; the number of orderings of n things. 0! = 1.\sum_{k=1}^{n} a_ksummation
Add a_k for k = 1 up to n.\prod_{k=1}^{n} a_kproduct
Multiply a_k for k = 1 up to n.\emptysetempty set, cardinality
The set with no elements; the number of elements of A.

Mistoqsijiet li n-nies jistaqsu

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.

Partijiet minn din il-paġna huma adattati minn Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Kondensat u mill-ġdid spjegat hawn; żbalji huma tagħna.

Aktar fil 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