1. 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

2. 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.

3. 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

4. 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?

5. 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:

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

7. 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

8. 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?

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

10. 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?

11. 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?

12. 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 different combinations of people.

13. Class work
349, # 1 – 21 all