1. Sullivan Algebra and Trigonometry: Section 14.2 Objectives of this Section Solve Counting Problems Using the Multiplication Principle Solve Counting Problems Using Permutations Solve Counting Problems Using Combinations Solve Counting Problems Using Permutations Involving n Non-Distinct Objects

2. Multiplication Principle of Counting If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in different ways.

3. If a license plate consists of a letter, then 5 numbers, how many different types of license plates are possible?

4. A permutation is an ordered arrangement of n distinct objects without repetitions. The symbol P(n, r) represents the number of permutations of n distinct objects, taken r at a time, where r < n. Example: How many ways can you arrange 7 books on a shelf? There are 7 choices for the first book, six choices for the second book (since the first was already picked), etc... So: (7)(6)(5)(4)(3)(2)(1) = 7! = 5040 ways

5. Number of Permutations of n Distinct Objects Taken r at a Time The number of different arrangements from selecting r objects from a set of n objects (r < n), in which 1. the n objects are distinct 2. once an object is used, it cannot be repeated 3. order is important is given by the formula

6. Suppose you were in charge of selecting four performers from a group of twelve to perform at a talent show. How many ordered arrangements of performers do you have to choose from? Since you are selecting 4 from 12, find P(12,4)

7. A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol C(n, r) represents the number of combinations of n distinct objects taken r at a time, where r < n.

8. Number of Combinations of n Distinct Objects Taken r at a Time The number of different arrangements from selecting r objects from a set of n objects (r < n), in which 1. the n objects are distinct 2. once an object is used, it cannot be repeated 3. order is not important is given by the formula

10. How many ways can you form a committee of three people from a group of 25? Since you are selecting 3 from 25 and order does not matter, we find: