Definition
General II ( x + Y ) ⁿⁿ ⁿ可 二用 二用 二用 二用 二用 二 为. Generalized II Theorem uses this result to a negative or non-integer power. At this time, the right is no longer a polynomial, but the number of infinity.
The binary coefficient is important to combine mathematics, because its meaning is from n piece of object, and select k The total number of methods is therefore also called . From the definition, the product of the n (1 + x ) is expanded, with any k item x And n - K 1 multiply a x , so the factor of x is from < I> nNumber of methods for selecting k . It can be more clearly seen about the x tag: When n = 4, K = 2,
The two-way coefficient is the No. 1 I> N + 1line of Yang Hui triangle from left to the left, i> k + 1 , which is first found by Yang Hui .
The two-term coefficients compliant equation can be certified by its formula or can also be derived from its meaning of combined mathematics. If the first left term represents the number of methods from K parts from n +1, these methods can be divided into No. I> n+ One piece, that is, select k pieces from the other n ; and select the n +1 pieces, that is, from the other N Pieces Select K -1. The second form is a method of selecting k from n , or it is also possible to select the rest of the n - K method of component.
Discovery History
The two-way factor table is called Jia Xian Triangle or Yang Hui triangle in my country, which is generally found to be the first in the Northern Song Mathematians. It is recorded in Yang Hui's "Detailed Nine Chapter Algorithm" (1261). A binomial positive factor table is also given in the work of Arab mathematician Cassi, and the calculation method he uses is exactly the same as Jia Xian.
In Europe, German mathematician Apiannus is engraved with this picture on the cover of the arithmetic book published in his 1527. But generally referred to as a Pasca triangle, because Pascal has also discovered this result in 1654. In any case, the discovery of the binary theorem is at least 300 years earlier than in Europe. In 1665, Newton promoted the binary theorem to N as the score and negative case, gave the expansion. The binary theorem has a wide range of applications in combination theory, high-level, high-order equivalence summation, and differential method.
Properties
Symmetry
is equal to two two-two-two-two-two-two-two factors of the first two "equal distances".
Single Peak
(1) When n is an even number, the middle one of the two-term factors
(2) When N is an odd number, the intermediate two-term factor
two-term coefficients and
binary theorem
two-term theorem ( Binomial Theorem, also known as Newton II, which is proposed by Aisak Newton on 1664,1665.
This theorem pointed out:
The I Item Item can be expressed as
binomial theorem, refers to the expansion of (A + B) ⁿ in N is positive integer.
Arrangement and combination
1,
2,
3 ,
as a = b = 1 When I enters the binary theorem, it can be proved that 1
When A = -1, b = 1, the induction 2-based theorem can prove 2
4. Composite Number :