## 12.1 Outline

- Election problem
- tree diagram
- fundamental counting principle
- arrangement
- ordered pair/ordered triple
- define permutation
- notation for permutations
- evaluating permutations

- Factorial
- definition
- multiplication property of factorials
- count-down property
- permutation formula

- Distinguishable permutations
- indistinguishable items
- distinguishable items
- 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 *n*_{1} of one kind, *n*_{2} of another, and*n* _{3} of still a third kind (so that *n*_{1}* + n*_{2}* + n*_{3}* =n*), then the number of ways that the *n* objects can be selected is *n!* divided by the product of (*n*_{1})!(*n*_{2})!(*n*_{3})!.

**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* ) !