If one event can occur in "m" ways, a second event in "n" ways and a third event in "r," then the three events can occur in "m\\times n\\times r" ways.
If one event with "n" outcomes occurs "r" times with repetition allowed, then the number of ordered arrangements is "n^r."
Distinctly ordered sets are called arrangements or permutations (order matters, no repetition).Â
The factorial is computed by the formula "n!=1\\cdot 2 \\cdot \\dots \\cdot n."
The number of permutations of size "k" that can be formed from the "n" individuals or objects in a group will be denoted by "P_{n,k}."
The number of permutations of "n" objects taken "k" at a time is given by:
If the "n" objects are all distinguishable there are "n!" permutations.
Dividing "n!" by "n_1!" gives the number of permutations of "n" objects with "n_1" of them being identical.
Repeating, to identify "n_2" objects of type 2, …, "n_k" objects of type "k," gives
as the result.
Example
If we have "n" elements of which "x" are alike of one kind, "y" are alike of another kind, "z" are alike of another kind, then the number of ordered selections or permutations is given by:
An unordered subset is called a combination. One way to denote the number of combinations is
"C_{n,k}."
Let "a_1,a_2,...,a_n" be "n" objects. A simple combination (or combination without repetition) of "k" objects from the "n" objects is one of the possible ways to form a set containing "k" of the "n" objects.
To form a valid set, any object can be chosen only once. Furthermore, the order in which the objects are chosen does not matter.
The difference between a multiset and a set is the following: the same object is allowed to appear more than once in the list of members of a multiset, while the same object is allowed to appear only once in the list of members of an ordinary set.
Let "a_1,a_2,...,a_n" be "n" objects. A combination with repetition of "k" objects from the "n" objects is one of the possible ways to form a multiset containing "k" of the "n" objects.
Where "n" is the number of things to choose from, and you choose "r" of them.Â
(Use permutation formulas when order matters in the problem.)
Where "n" is the number of things to choose from, and you choose "k" of them.Â
(Use combination formulas when order doesn’t matter in the problem.)
Where "n" is the number of things to choose from, and you choose "k" of them.Â
(Use combination formulas when order doesn’t matter in the problem.)
Where "n" is the number of things to choose from, and you choose "k" of them.
There are
"{n! \\over n_1!n_2!...n_k!}"ways to put "n" distinguishable objects into "k" boxes, so that the "i"th box contains "n_i" objects.
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