Factorial
https://mathworld.wolfram.com/Factorial.html
The factorial is defined for a positive integer as
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(1)
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So, for example, .
Permutation
https://mathworld.wolfram.com/Permutation.html
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
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(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.
Combination
https://mathworld.wolfram.com/Combination.html
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
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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
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(2)
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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