Essential Ideas for Section 12.3

Fundamental Counting Principle

If one task can be performed in m ways, and if, after that task is completed, a second task can be performed in n ways, then the total number of ways of ways for both tasks is found by multiplication: mn. Repetitions are allowed.

Permutations

A permutation of r elements selected from a set of n elements is an ordered arrangement of those r elements selected without repetions. The order of selection is important.

Combinations

A combination of r elements selected from a set of n elements is an subset of r elements selected without repetions. The order of selection is not important.

Counting Formulas

Factorial: n! = n(n - 1)(n - 2)(n - 3) ... (3)(2)(1).


Count-down formula: n! = n(n - 1)!


Permutation formula: The number of ways of selecting r elements from a set with cardinality n in which the order of section is important is n!/(n - r)!

Combination formula: The number of ways of selecting r elements from a set with cardinality n in which the order of selection is not important is n!/r!(n - r)!.

Number of distinguishible permutations:
If n objects are partitioned so that there are n1 of one kind, n2 of another, and n3 of still a third kind (so that n1 + n2 + n3 = n), then the number of ways that the n objects can be selected is n! divided by the product of (n1)!(n2)!(n3)!.