The proof of this formula can be done by mathematical induction or multinomial theorem. In multinomial theorem, we can regard (a+b) as a polynomial, and then use multinomial theorem to get its expansion, so as to get the formula of binomial theorem.
The binomial theorem has some properties and changes. For example, when b is equal to 1, the binomial theorem becomes Pascal triangle. When both a and b are equal to 1, the binomial theorem becomes the form of Bernoulli number. These changes and properties further expand the application scope and manifestation of binomial theorem.
Binomial theorem is a basic mathematical theorem, which describes the coefficient law of a given power series expansion. This theorem can be used to solve many mathematical problems, including combinatorial mathematics, algebra, probability theory and other fields. The binomial theorem was originally used to open higher powers. 1654, Pascal of France first established the binomial theorem of general positive integer power, so the arithmetic triangle is still named after him in the west.
Application of binomial theorem:
Combinatorial mathematics: binomial theorem can be used to calculate the number of combinations and permutations. In combinatorial mathematics, binomial theorem is used to calculate the number of combinations of k elements selected from n different elements, or to arrange n elements in k different positions. This is the most basic application of binomial theorem in combinatorial mathematics.
2. Algebra: In algebra, binomial theorem is used to expand a polynomial or solve an equation. For example, (a+b) n can be expanded into an n+c (n,1) a (n-1) b+...+c (n, n- 1) ab (n-65438). This application scenario is binomial theorem in algebra.
3. Probability theory: In probability theory, binomial theorem can be used to calculate the probability or expected value of some events. For example, in Bernoulli's experiment, binomial theorem can be used to calculate the probability that event A happens exactly k times. In addition, binomial theorem can also be used to calculate variance, covariance and correlation coefficient of discrete random variables.
4. Calculus: In calculus, binomial theorem can be used to approximately calculate the values of some functions. For example, sin(x) can be expanded into Taylor series by binomial theorem. In addition, binomial theorem can also be used to solve approximate solutions of some differential equations.
5. Physics: In physics, binomial theorem can be used to describe the solution of wave function and angular momentum in quantum mechanics. In addition, binomial theorem can also be used to calculate the approximation of some physical quantities, such as the trajectory of planets.
6. Computer science: In computer science, binomial theorem can be used to optimize algorithms and improve computational efficiency. For example, binomial theorem can be used to calculate factorial and power operation quickly. In addition, binomial theorem can also be used to design some data structures and algorithms, such as quick sorting and merge sorting.