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

ປັດຈຸບັນ​ທ່ານ ບໍ່ມີ​ເຄື່ອງ​ຄິດໄລ່​ທີ່​ຈະ​ແກ້ໄຂ​ບັນຫາ​ນີ້​ໄດ້, ແຕ່​ສ່ວນ​ຂອງ​ມັນ​ສາມາດ​ຄິດໄລ່​ໄດ້. ພະຍາຍາມ​ອັນ​ໃດ​ອັນ​ໜຶ່ງ​ຂ້າງ​ລຸ່ມນີ້ ຫຼື ພິມ​ຕົວ​ເອງ​ເອງ.

⌨ ປະເພດ ✎ ແຕ້ມມັນ

ບັນຫາ​ຄະນິດສາດ​ຂອງ​ທ່ານ

π ແກ້ໄຂ

ທົດລອງ (1)

ທົດລອງ​ທຸກ​ຢ່າງ​ກ່ອນ​ໜ້າ​ນີ້​ໃນ​ເຈ້ຍ. ເປີດ​ເຜີຍ​ຄໍາຕອບ​ເພື່ອ​ກວດ​ສອບ; ຄໍາຕອບ​ທີ່​ໄດ້​ຮັບ​ການ​ກວດ​ສອບ​ສາມາດ​ເປີດ​ໃນ​ຕົວ​ແກ້​ໄຂ​ສຳລັບ​ທຸກ​ບາດກ້າວ.

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

   ເຮັດວຽກຜ່ານມັນ ຂໍ້​ແນະ​ນຳ

ຮັກສາ​ການ​ເຮັດວຽກ​ຂອງ​ທ່ານ​ໄວ້

ບັນຊີຟຣີ ເພີ່ມ​ຂໍ້​ມູນ​ໃສ່​ບົດຮຽນ​ທຸກ​ບົດ, ບັນທຶກ​ຂອງ​ສິ່ງທີ່​ທ່ານ​ໄດ້​ເຮັດ​ແລ້ວ, ບັນຫາ​ທີ່​ທ່ານ​ໄດ້​ແກ້​ໄຂ​ໄວ້​ໃນ​ບ່ອນ​ດຽວ ແລະ ຄູ​ສອນ​ທີ່​ທ່ານ​ສາມາດ​ຖາມ​ກ່ຽວກັບ​ໜ້າ​ນີ້. ຄະນິດສາດ​ເອງ​ແມ່ນ​ເປີດ​ໃຫ້​ທຸກ​ຄົນ, ບໍ່​ວ່າ​ຈະ​ໄດ້​ເຂົ້າ​ສູ່​ລະບົບ ຫຼື ບໍ່.

ລົງທະບຽນ ចូល

ຕົວ​ສະ​ແດງ​ທີ່​ໃຊ້​ຢູ່​ທີ່​ນີ້

ກົດ​ຕົວ​ອັກສອນ​ໃດ​ໜຶ່ງ​ເພື່ອ​ເບິ່ງ​ຄວາມ​ໝາຍ​ເຕັມ, ຮູບ ແລະ ຕົວອັກສອນ​ແຕ່ລະ​ຕົວ​ໃນ​ມັນ​ໝາຍ​ຄວາມ​ວ່າ​ແນວໃດ.

\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.

ຄໍາຖາມທີ່ຄົນຖາມ

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.

ບາງສ່ວນຂອງ ໜ້າ ນີ້ ຖືກປັບແຕ່ງຈາກ Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). ຖືກ​ປະສົມ​ປະສານ​ແລະ​ອະທິບາຍ​ຄືນ​ໃໝ່​ທີ່​ນີ້; ຄວາມ​ຜິດ​ພາດ​ແມ່ນ​ຂອງ​ເຮົາ.

ເພີ່ມເຕີມໃນ 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