Permutations and combinations in mathematics is one area where factorial is extensively used. Other areas are probability, exponentials, number theory, calculus and physics.
i. Permutations and combinations
a. Linear permutations:
Consider arrangements of a fixed length of elements (say k) taken from a set of 'n' elements.
The numbers of such k-permutations of 'n' is given by
n!/(n – k)!
b. Circular permutations:
There is no 'first element' in such an arrangement. Any element can be considered the 'first element'.
The number of circular permutations of a set S with 'n' elements is given by
(n – 1)!
c. Multiset permutations :
It is an ordered arrangement of the 'n' elements of set M in which each element appears the same number of times as in M.
A good example of multiset permutation is the anagrams of a word having repeated letters.
Say a word "W" contains 'n' letters, out of which 3 letters are repeating m1, m2 and m3 times. (Their order is not important).
Then the number of multiset permutations of W is given by.
d. Combinations:
It is also a selection of items from a given set. But unlike permutations, here the order of selection of items does not matter.
If the set has 'n' elements, the number of ways of selecting 'r' elements is given by
nCr = 