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

Ugbu a ị Enweghị kalkulata na-ewepụ nke a, ma ọ bụ na ọ nwere ike ijikwa ya. Jiri otu n'okpuru, mọọbụ pịa nke gị.

⌨ Ụdị ✎ Debata ya

Nsogbu igodo gị

π Wepụ

Nhazi (1)

Nwapụta n'ime akwụkwọ n'oge mbụ. Gosi nzaghachi ka ịlelee ya; ndị ahụ e gosipụtara nwere ike ịbụ ndị a mepere n'ime onye nrụpụta maka nzọụkwụ ọbụla.

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

   Ọrụ site na ya Ndụ

Kwụsị ọrụ gị

Akwụkwọ ndebanye n'efu na-egbakwunye ntọala n'ụdị ọmụmụ ihe ọbụla, ndepụta nke ihe ị ga-eme, nsogbu ị ga-ewepụ n'otu ebe, na onye nkụzi ị ga-ajụ banyere ihuakwụkwọ a. Akwụkwọ ndepụta ihe bụ nke ọbụla mepere maka onye ọbụla, ọbụla ma ọ bụ na ọ bụghị.

Akaụntụ Nbanye

Aịkọn ndị ahụ ejirila ebe a

Tinye ọbụla simbụl maka nkọwa zuru ezu, inyogo, na ihe ọbụla ọbụla na-egosi.

\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.

Ajụjụ ndị mmadụ jụrụ

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.

Oge nke ihuakwụkwọ a bụ nke a gbanwere site na Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). E kọwaara ya na ebe a; nsogbu bụ nke anyị.

Oge 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