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7.2 Pascal’s Triangle and Combinations

7.2 Pascal’s Triangle and Combinations. 4/10/2013. In today’s lesson we’re learning…. h ow to find the possible number of combinations given a situation and how it relates to Pascal’s triangle. Factorial !. Definition:. The product of an integer and all the integers below it. 0! = 1

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7.2 Pascal’s Triangle and Combinations

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

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

  5. Example: Ice creamThere 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. nCkFormula nCk= For the ice cream example: 4C2 “4 choose 2” since there are 4 flavors and you get to choose 2. 4C2 = = 6

  7. Find the number of combinations: nCk= 1. 6C2 6C2 = 2. 7C4 7C4= = 15 = 35

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

  9. Now let’s do some word problems! 3 types of problems and what to do. • “exactly” – multiply each group • “at least” – add each group. • “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? 6C2 for veggies 4C3 for meat “exactly” multiply each group. 6C2•4C3 = 15• 4 = 60 4C3 6C2

  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? What we have is this: 18C3 Since our Pascal’s Triangle is not big enough to show the 18th row, let’s use the Combination formula. nCk = 18C3= = 816

  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! 12C9 + 12C10 + 12C11 + 12C12 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! 6C0 + 6C1+ 6C2+ 6C3+ 6C4 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!!!

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