7.2 Pascal’s Triangle and Combinations

4/10/2013

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.

Factorial !The product of an integer and all the integers below it.0! = 11! = 12! = 2•1 =23! = 3•2•1 = 64!= 4•3•2•1 = 24

Definition:

How to Calculate:

Combinationis 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. nCk is the number of possible combinations from that choice.

Definition:

How it is written:

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 creamRocky RoadVanillaMint ChipStrawberry

List of possible combinations:RVRMRSVMVSMSThere are 6 combinations.

Luckily, there’s a formula you can use instead of making a list!!! Cool huh?

nCk Formula

nCk = For the ice cream example: 4C2 “4 choose 2” since there are 4 flavors and you get to choose 2.

4C2 = = 6

Find the number of combinations:nCk =

1. 6C2

6C2 =

2. 7C4

7C4 =

= 15

= 35

So how does this relate to Pascal’s Triangle?

Note: The numbers in the Pascal’s Triangle represents nCk

Now let’s do some word problems!3 types of problems and what to do.1. “exactly” – multiply each group2. “at least” – add each group.3. “at most” – add each group.

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

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 = ¿18 ∙17 ∙16 ∙15… .3 ∙2∙115 ∙14 ∙13. .3 ∙2∙1∙3 ∙2 ∙1

= 816

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

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

Homework

WS 7.2Skip #s 8, 9, 12 and 14.

What does a clock do when it gets hungry???

It goes back four seconds!!!