# Section 12.1: Permutations

## 12.1 Outline

1. Election problem
1. tree diagram
2. fundamental counting principle
3. arrangement
4. ordered pair/ordered triple
5. define permutation
6. notation for permutations
7. evaluating permutations
2. Factorial
1. definition
2. multiplication property of factorials
3. count-down property
4. permutation formula
3. Distinguishable permutations
1. indistinguishable items
2. distinguishable items
3. formula for distinguishable permutations

## 12.1 Essential Ideas

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.

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

Number of distinguishable permutations: If n objects are partitioned so that there are n1 of one kind, n2 of another, andn 3 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)!.

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 ) !