SOLUTION: What is the formula for combinatorics and permutations?

Question 198820: What is the formula for combinatorics and permutations?
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Combinatorics is a branch of pure mathematics concerning the study of discrete (and usually finite) objects. It is related to many other areas of mathematics, such as algebra, probability theory, ergodic theory and geometry, as well as to applied subjects in computer science and statistical physics. Aspects of combinatorics include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and finding algebraic structures these objects may have (algebraic combinatorics).
[Source: http://en.wikipedia.org/wiki/Combinatorics]

So there is no "formula for combinatorics". There is, however, a formula for combinations. Don't you DARE roll your eyes and say "Whatever" either. Mathematics is a very precise science and therefore requires exquisite precision of language when describing it.

The number of combinations of

things taken

at a time is:

Also denoted

The number of permutations of

things taken

at a time is:

Also denoted

Notice that the difference is the factor of

in the denominator of the formula for combinations. That factor represents the number of ways that

things can be ordered, hence use the formula for combinations when order DOES NOT matter, and permutations when order DOES matter. This fact also gives rise to what may become a handy relationship to know:

John

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