Maths: Factorial 阶乘 / Permutation 排列 / Combination 组合 / Binomial Theorem 二项式定理

A permutation, also called an "arrangement number" or "order," is a rearrangement of the elements of an ordered list  into a one-to-one correspondence with  itself.

The number of ways of obtaining an ordered subset of  elements from a set of  elements is given by

(1)

(Uspensky 1937, p. 18), where  is a factorial. For example, there are  2-subsets of , namely , , , , , , , , , , , and . The unordered subsets containing  elements are known as the k-subsets of a given set.

The number of ways of picking  unordered outcomes from  possibilities. Also known as the binomial coefficient or choice number and read " choose ,"

where  is a factorial (Uspensky 1937, p. 18). For example, there are  combinations of two elements out of the set , namely , , , , , and . These combinations are known as k-subsets.

Binomial Theorem

https://mathworld.wolfram.com/BinomialTheorem.html

The most general case of the binomial theorem is the binomial series identity

(1)

where  is a binomial coefficient and  is a real number. This series converges for  an integer, or . This general form is what Graham et al. (1994, p. 162). Arfken (1985, p. 307) calls the special case of this formula with  the binomial theorem.

When  is a positive integer , the series terminates at  and can be written in the form

(2)

This form of the identity is called the binomial theorem by Abramowitz and Stegun (1972, p. 10).

更多参考：

https://www.mathsisfun.com/combinatorics/combinations-permutations.html