Factorial
https://mathworld.wolfram.com/Factorial.html
The factorial
is defined for a positive integer
as
 |
(1)
|
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
 |
(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.
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
 |
(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