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

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⌨ Jenis ✎ Lukis

Masalah matematik awak.

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Latihan (1)

Cuba setiap satu di atas kertas dahulu. Tunjukkan jawapan untuk semak; yang disahkan boleh dibuka dalam penyelesai bagi setiap langkah.

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

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Akaun percuma menambah nota pada setiap pengajaran, rekod apa yang anda selesaikan, masalah yang anda selesaikan di satu tempat, dan seorang guru yang anda boleh tanya tentang halaman ini. Matematik itu sendiri terbuka kepada semua orang, sama ada anda log masuk atau tidak.

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Simbol yang digunakan di sini

Tekan simbol mana-mana untuk definisi penuh, gambar, dan apa makna setiap huruf dalamnya.

\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.

Soalan yang orang tanya

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.

Bahagian halaman ini diadaptasi dari Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). Dikemas kini dan dijelaskan di sini; kesilapan adalah kami.

Lebih dalam 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