1. Multiplication Counting Principle How many ways can you make an outfit out of 2 shirts and 4 pants? If there are m choices for step 1 and n choices for step 2, then the total number of choices for both steps is m * n Example: How many license plates can be made with 3 letters followed by 2 digits then a letter? 26 X 26 X 26 X 10 X 10 X 26 Example: A pizza shop offers 3 types of crust and 8 toppings. How many unique 1-topping pizzas could be ordered? 3 X 8 = 24 Example: How many different ice cream sundaes could be made with a choice from 3 flavors of ice cream, 2 syrups, and 3 toppings. 3 X 2 X 3 = 18 A combo meal consists of a sandwich, side, and drink. How many meals could be made from choices of 4 sandwiches, 6 sides, and 3 drinks? 4 X 6 X 3 =72

2. In the US, TV and radio stations are identified by a 4 letter code. In the East, the code starts with W. In the West, the code starts with K. How many different station names can be made? 2 X 26 X 26 X 26 How many ways can you create a schedule out of 7 different classes that you want to take? 7 X 6 X 5 X 4 X 3 X 2 X 1 = 5040 Draw a tree diagram showing the ways that you can flip a coin And spin a 4-section spinner. How many different outcomes are there? What is the probability of flipping HEADS and spinning a 3?

3. PERMUTATIONS The number of ways to put n items IN ORDER Example: There are 12 songs on a digital album. If you use the Shuffle option, how many different orders can the songs be played? 1211 109 8 7 6 5 4 3 2 1 MULTIPLY to get the answer Example: 9 baseball players start a game. How many different Batting orders can be made? 987 6 5 4 3 2 1MULTIPLY to get the answer Example:You have 5 errands to run but only have time for 3 at lunch. How many different ways can you get those done? 5 43MULTIPLY to get 60 To Order n items, the total is n! To Order r out of n items, the total is nPr = n! (n-r)! Example: A club with 10 members is electing Pres, VP, Sec & Treas. How many different ways can those officers be chosen? 10 P 4 = 10! 6! OR 10 X 9 X 8 X 7 = 5040

4. List all the ways to arrange the letters in the word CAT List all the ways to arrange the letters in the word BEE List all the ways to arrange the letters in the word LION List all the ways to arrange the letters in the word DEER Permutations with Repeats: To count the number of ways to arrange a group of n items containing r duplicate copies of an item, s duplicate items of another item, etc. n! r! s!... Example: How many ways can the letters in the word STATISTICS be arranged? S : 3 T : 3 I : 2 10! (3! 3! 2!) 3628800 72 50,400 How many ways can 7 flags be arranged if there are 2 blue, 2 red, and 3yellow? 7! (2! 2! 3! ) 210

5. Combinations The number of ways to make groups of size r out of a total of n items nCr = n! r!(n-r)! Ex. Handshakes are groups of size 2. nC 2 = ORDER DOESN’T MATTER. How many ways can 4 representatives be chosen from a group of 10 students? How many ways can a teacher pick 4 books to read with English class out of 8 choices in the curriculum? If you have 4 favorite books, what is the probability that the teacher chooses your 4?

6. Combinations with subgroups When making combinations (groups) sometimes there are smaller Groups that must be accounted for. “AND” : multiply “OR” : add Ex. A class is made up of 10 boys and 13 girls. How many different Groups of 6 students can be formed if 3 are boys and 3 are girls? (Boys) (Girls) = (10C3 ) (13C3) 120 * 286 = 34,320 Ex. A basketball team is made up of 5 Seniors and 7 Juniors. How many starting teams can be chosen that are made up of exactly 3 Seniors and 2 Juniors? (Seniors)(Juniors) =(5C3)(7C2)10 * 21 = 210 How many total starting teams of 5 can be made? 12C5 = 792 What is the PROBABILITY that the starters are 3 Seniors and 2 Juniors? 210 / 792 0.265 How many ways can a starting team be set up with AT LEAST 3 Seniors? (5C3)(7C2) +(5C4)(7C1) + (5C5)(7C0)