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

Indi sen Hiç bir hasapçy buny çözmez, emma onuň bölekleri hasaplanmak üçindir. Aşakda birini synla ýa-da özüňi ýaz.

⌨ _Täze ✎ Ony çyz

Siziň rime problemaňyz

π Çöl

Tälim (1)

Her birini kagyzda synlap gör. Çözgüni barlap görmek üçin aç; barlap görlenler her bir ädim üçin çözgüççide açylyp bilner.

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

   _Işle Eşik

Özüň işle

Mugt hasap her bir ders üçin notlar goşýar, tamamlanan zadyňyzyň ýazgysyny, çözen problemaňyzy bir ýerde, we bu sahypa barada sorap biljek mugallymy. Matematika özüçe her kim üçin açyk, girseň ýa girmeseň.

_Gün _Giriş

Şu ýerde ulanylan simwollar

Eň täze düşündirme, surat, we her harpyň manysy üçin her bir simwoly basyň.

\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.

Adamlar soraýan soraglar

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.

Bu sahypanyň käbir bölegi şundan alyndy Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Bu ýerde gysgaldylan we tekrar beýan edilen; hatalar biziňkidir.

_Ýaşa 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