## 12.3 Outline

- Distinguish permutations and combinations
- License plate problem
- Which method?

## 12.3 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*. 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 repetitions*. 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 repetitions*. 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 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})!.