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

Haddaad Ma jiro xisaabis oo ku nool midkan, laakiin qaybaha ka mid ah waa la xisaabi karaa. Ku raaxayso mid ka hooseeya, ama ku qor adiga kuu gaar ah.

⌨ Nooc ✎ Sawir

Su'aalkaaga xisaabta

π Xalin

Tababar (1)

Ku day mid kasta oo ka mid ah oo ku saabsan buugaag hore. Muuji jawaabta si aad u hubiso; kuwa la xaqiijiyay waxaa la furi karaa in xalka ah tallaabo kasta.

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

   Shaqada iyada oo Tallaabo

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

\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.

Su'aalaha dadku waydiiyaan

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.

Qaybaha ee boggaan waxaa la beddelay ka Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Condensed iyo dib-oo halkan lagu sharaxay; khaladaadka waa inoo.

In ka badan 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