The Fundamental Counting Principle and Permutations

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

3. Let’s have a little fun! http://www.classzone.com/cz/books/algebra_2_cs/resources/applications/animations/html/alg207_ch10_pg689.html

4. Fundamental Counting Principle Two Events: • If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is Three Events: • If one event can occur in m ways, a second event in n ways, and a third event in p ways, then the number of ways that all three events can occur is More Events: • How will the formula look like for four events? • How about five events? For Two Events For Three Events More events

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

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

7. Checkpoint 2. 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? 10,000 PINs; 5040 PINs ANSWER Use the Fundamental Counting Principle

8. Permutations of n Objects The number of permutations of n distinct objects is n! Examples Solutions: • 6!=6*5*4*3*2*1 • 5!=5*4*3*2*1 • 4!=4*3*2*1 • 0!=1 Examples: 6 objects 5 objects 4 objects 0 objects

9. Permutations of n Objects Taken r at a Time The number of permutations of n objects taken r at a time is denoted by and is given by the following formula: Examples Solutions: • 360 • 720 Examples: You have 6 text messages on your cell phone. How many orders can you reply to 4 of the messages? To all 6 of the messages?

10. Example 4 SOLUTION Find the number of permutations of 6 objects taken 4 at a time. 6! 6! 6 • 5 • 4 • 3 • 2 • 1 6P4 360 = = = = = ( ) 6 – 4 ! 2! 6 • 5 • 4 • 3 2 • 1 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?

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

12. Conclusions Summary Assignment Pg 542 #(20,26,35,40,44,50,55,59) Exit Slip: • How do you determine the number of distinguishable permutations of n objects taken r at a time? • You can use the formula .