The document discusses various counting methods including listing elements, tree diagrams, and the fundamental counting principle. It also covers permutations and combinations. Specifically, it defines a permutation as an ordering of distinct objects and gives the formula P(n,r) = n!/(n-r) to calculate the number of permutations. It further defines a combination as a set of objects where order does not matter.

1. Counting & Bases Chapter 12 & Chapter 4.3 BY: Gessica, Crystal, Israel, and Quentin

2. Counting Methods We can count a set by listing its elements systematically. We can use tree diagrams to help visualize counting situations. We can solve counting problems by imagining a tree instead of drawing it

3. Counting sets by listing One of the simplest ways to count a set is to list its elements. Examples: A. Flip a coin a) two ways: Heads or Tails.

4. Tree Diagram Tree diagram is handy way to illustrate the possibilities. How many ways can three coins be flipped??

5. … cont’d Forming two letter words using D and B. how many ways can it be formed? Depends if repetition is allowed or not.

6. Dice Set To emphasize that the dice are different, we assume that one is purple and the other is green. Clearly a 2red and a 3green is not the same as a 2green and 3red. We use ordered pairs There are 36 pairs from a pair of dice

7. The Fundamental Counting Principle If we wish to perform a series of task and the first task can be done in A ways, second B ways, and third C ways.. Task can be done in A x B x C ways.

8. FCP in Coin and Dice Problems. How many ways can a four coins be flipped? - two ways possible. 2 x 2 x 2 x 2 = 16 ways! How many ways can three dice (red, blue, green) be rolled? -rolling die can be done 6 ways. 6 x 6 x 6 = 216 ways!

9. Permutation A permutation is an ordering of distinct objects in a straight line. P(n, r) : given “n” objects and selecting “r” of them. Formula for computing P(n, r) - P(n, r) = n! Divided by (n – r)

10. combination A combination is a set of objects in which order is not imporant

11. Factorial Notation! Factorial notation is often used in questions in combinatorics. It is when you need to multiply consecutive numbers. When you see x ! it is the factorial of x that is being asked for. When asked to calculate the following 7! your solution will look something like: 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040. Your calculator will have a key that will perform this operation for you.