1. The Fundamental Counting Principle and Permutations 18.0 Students use fundamental counting principles to compute combinations and permutations. 19.0 Students use combinations and permutations to compute probabilities.

2. The Fundamental Counting Principle and Permutations Objectives Use the fundamental counting principle and permutations. – Fundamental Counting Principle – Permutations of n Objects – Permutations of n Objects Taken r at a Time Use the fundamental counting principle and permutations. – Fundamental Counting Principle – Permutations of n Objects – Permutations of n Objects Taken r at a Time Key Words Permutation – An ordering of a set of objects Factorial – The number of permutations of n distinct objects is n! Permutation – An ordering of a set of objects Factorial – The number of permutations of n distinct objects is n!

3. Let’s have a little fun! http://www.classzone.com/cz/books/algebra_2_ cs/resources/applications/animations/html/alg2 07_ch10_pg689.html

4. Fundamental Counting Principle 1.For Two Events 2.For Three Events 3.More events

5. Example 2 Use the Fundamental Counting Principle Passwords You are choosing a password that has 4 letters followed by 2 digits. N C W J 3 7 4 letters 2 digits How many passwords are possible if letters and digits can be repeated? ANSWER The number of different passwords is 45,697,600. SOLUTION There are 26 choices for each letter and 10 choices for each digit. Use the fundamental counting principle. = 45,697,600 Number of passwords = 26 26 26 10 10

6. Checkpoint Use the Fundamental Counting Principle Suppose that the lunch special also comes with your choice of 3 desserts. How many lunch specials are possible? 1. ANSWER 18 lunch specials

7. Checkpoint Use the Fundamental Counting Principle You are choosing a personal identification number (PIN) for your ATM card. The PIN has 4 digits. How many PINs are possible if digits can be repeated? if digits cannot be repeated? 2. ANSWER 10,000 PINs; 5040 PINs

8. Permutations of n Objects Examples: 1.6 objects 2.5 objects 3.4 objects 4.0 objects

9. Permutations of n Objects Taken r at a Time Examples: You have 6 text messages on your cell phone. 1.How many orders can you reply to 4 of the messages? 2.To all 6 of the messages?

10. Example 4 Permutations of n Objects Taken r at a Time Text Messages You have 6 text messages on your cell phone. In how many orders can you reply to 4 of the messages? to all 6 of the messages? SOLUTION Find the number of permutations of 6 objects taken 4 at a time. 6 P46 P4 = 64–! 6!6! = 6! 2! == 360 () 654321 21 = 6543

11. Checkpoint Find Permutations In how many different orders can 12 snowboarders finish a competition? 3. In how many orders can you respond to 5 of 8 text messages? 4. ANSWER 479,001,600 orders ANSWER 6720 orders

12. Conclusions Summary Assignment Pg 542 – #(20,26,35,40,44,50,55,59) Pg 542 – #(20,26,35,40,44,50,55,59)