1. 7.2 Pascal’s Triangle and Combinations 4/10/2013

2. In today’s lesson we’re learning… how to find the possible number of combinations given a situation and how it relates to Pascal’s triangle.

3. Factorial ! The product of an integer and all the integers below it. 0! = 1 1! = 1 2! = 21 =2 3! = 321 = 6 4!= 4321 = 24 Definition: How to Calculate:

4. Combination is a way of selecting several things out of a larger group, where order does not matter. nCknCk read as “n Choose k”. That means that you have n number of selections and you’re choosing k amount. n C k is the number of possible combinations from that choice. Definition: How it is written:

5. Example: Ice cream There are 4 flavors of ice cream you can choose from and you get to pick 2. How many 2-flavor combinations can you have? 4 flavors of ice cream Rocky Road Vanilla Mint Chip Strawberry List of possible combinations: RV RM RS VM VS MS There are 6 combinations. Luckily, there’s a formula you can use instead of making a list!!! Cool huh?

6. n C k Formula = 6

7. Find the number of combinations: = 15 = 35

8. So how does this relate to Pascal’s Triangle? Note: The numbers in the Pascal’s Triangle represents n C k

9. Now let’s do some word problems! 3 types of problems and what to do. 1.“exactly” – multiply each group 2.“at least” – add each group. 3.“at most” – add each group.

10. A restaurant gives options of 6 vegetables and 4 meats be ordered in an omelet. Suppose you want exactly 2 vegetables and 3 meats in your omelet. How different omelets can you order? 6 C 2 for veggies 4 C 3 for meat “exactly” multiply each group. 6 C 2 4 C 3 = 15 4 = 60 4C34C3 6C26C2

11. You are going to buy a bouquet of flowers. The florist has 18 different types of flowers. You want exactly 3 types of flowers. How many different combinations of flowers can you use in your bouquet?

12. During the school year, the basketball team is scheduled to play 12 home games. You want to attend at least 9 of the games. How many different combinations of games can you attend? At least 9 games means you can attend 9, 10, 11, 12. So ADD all the possibilities! 12 C 9 + 12 C 10 + 12 C 11 + 12 C 12 220 + 66 + 12 + 1 = 299

13. You only like 6 songs on the latest Arcade Fire album. If you want to purchase at most 4 songs with the credit you have on iTunes, how many different combinations can you buy? At most 4 songs means you can buy 0, 1, 2, 3 or 4. So ADD all the possibilities! 6 C 0 + 6 C 1 + 6 C 2 + 6 C 3 + 6 C 4 1 + 6 + 15 + 20 +15 = 57

14. Homework WS 7.2 Skip #s 8, 9, 12 and 14. What does a clock do when it gets hungry??? It goes back four seconds!!!