1. Permutations fancy word isn’t it…

2. Making Groups I need 4 volunteers We are going to make groups of 3 people Let’s make those groups What situations make the order important? How did you organize the information? Come up to the board and write how you recorded your information.

3. Fitting objects into places You run a tanker company You have n items and r places to put them Determine the total number of ways that you can place these items into those places Create a chart like the following: 12345 1 2 3 4 5 n = number of objects r = number of spaces 12 6 26 1220 3 45 2460 24 120 Does anyone see any patterns?

4. Think about our tree diagrams Let’s look at the 4 pieces (r) and 3 places (n) How many branches would you create for your first space to have something placed into it? How many branches would come off of each initial branch? Why is it less each time? A B C DA First Space Second Space Third Space A B C D

5. 3 spaces, 7 objects How many different arrangements could we make from this? 7 x 6 x 5 210 Since we have 7 possibilities for the first spot, 6 in the following spot and 5 in the final spot You could also draw a tree – 7 branches for the first area, 6 off of each of those, 5 off of each of the 6… This is called the counting principle

6. Factorial So if we had 7 spaces and 7 objects, how many arrangements would we have possible? 7 x 6 x 5 x 4 x 3 x 2 x 1 We can write this as 7! This reads: “7 factorial” We can do this operation on our calculators MATH  PRB  4

7. Alphabet soup We have a can of alphabet soup and we have one of each letter. You are going to make a word with ALL 13 letters in the soup. (They don’t have 26 letters) How many different words would you have? 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 Which is the same as 13! Using a calculator: 6227020800 different options

8. What if we don’t use all of 13! If we had 13 letters but only wanted to make words with 9 letters. What operation would I do? 13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 259459200 We have a shorter way of doing this operation =13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5

9. Notation for Permutations We have a formal way of writing this: For n pieces and r spaces, n P r = So for our case it would be: 13 P 9 = “The number of permutations of n things chosen r at a time”

10. Practice Solve for the following. Write it out using factorials then all of the steps to be multiplied, for example: 10 P 3 = 7 P 2 5 P 4 12 P 5 = 10 x 9 x 8 = 42 = 120 = 95040

11. Finding Probabilities Here is where it gets fun…I have one of each of the letters of Alphabet Soup in my bowl What is the probability of spelling BAT from my bowl if I randomly select 3 letters? So just like always, to find a probability you find the following: So we have one outcome of the total number of outcomes, 1716 P(BAT) = 1/1716 Number of desired outcomes Total number of outcomes

12. License Plates How many different license plates can the state of Colorado produce if the first three slots are numbers, and the last three slots are letters? What is the probability of getting 000 AAA? What is the probability of getting a plate that does not repeat? What is the probability of getting a James Bond (007) license plate? – saw one the other day

13. The Band There is a group performing tonight and the 7 musicians performing have their names randomly printed on the program. What is the probability that the list is in alphabetical order? If we only have four chairs for the performers to sit in, how many different ways can they seat themselves?

14. Combination Locks You are a lock manufacturer and are designing locks. How many different combinations are available for a lock with 3 different numbers if the numbers do not repeat? With 4 different numbers? How many more times “safe” is a 4 number lock than a 3 number lock?

15. Homework and Reading Read page 303 thru 304 Read the strategy used on page 305 p306-7 #4, 12a,b, 16-20, 24-26