Warm Up – Please list: (Please use tree diagrams, and then list the events) How many ways can we arrange 3 objects A, B, and C: Using just two How many ways can we arrange 4 objects, A, B, C, & D: Using only two Using only three Keep this. We will get back to this later today! 6 12 24

Math I UNIT QUESTION: How do we determine the number of options if order does not matter? Standard: MM1D1.b Today’s Question: How can we find the number of ways to select 9 people out of 20 for a party? Standard: MM1D1.b.

Combinations What if order does not matter? Let’s go back to the warm-up: How many ways can we organize 3 things, using just two of them, if order does matter? How many redundancies do we have if order does not matter? How may ways can we organize 3 things if order does not matter? How can we represent this? 3!/(number of redundancies) 6 2 3

How many ways can we select 2 things from 4 unique things if order does matter? How many ways can we select 2 things from 4 unique things if order does not matter? Make a tree diagram. Answer: 6 Note that we have 2 variations for every choice we pick, which can be written as 2! How can we represent this?

Combinations A combination is a selection of r object from a group of n objects without regard to order and is denoted by The number of combinations of r objects selected from a group of n objects is:

Example 1 How many different ways are there to select two class representatives from a class of 20 students?

Solution The answer is given by the number of 2-combinations of a set with 20 elements. This is often stated as 20 elements, taking 2 at a time The number of such combinations is

Example 2 From your class of 24, the teacher is randomly selecting 3 to help Mr. Boland with a project. How many combinations are possible?

Your turn! For your school pictures, you can choose 4 backgrounds from a list of 10. How many combinations of backgrounds are possible?

Your turn! Coach Hill randomly selects 3 people out of his class of 20 to go to the courts and help him get ready for a tennis match. How many possibilities of people does he have?

Permutations & Combinations Review How many ways can we select 5 objects from a group of 20? Does order matter? If it does (maybe we are filling specific positions on a team), we have a permutation problem. How many? If order does not matter (maybe we just want 5 different pieces of fruit) we have a combination problem. How many?

There are 167960 different combinations of people. Math I Today’s Question: How can we find the number of ways to select 9 people out of 20 for a party? There are 167960 different combinations of people.

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