2. 5.4 Counting Methods Objectives: By the end of this section, I will be able to… 1) Apply the Multiplication Rule for Counting to solve certain counting problems. 2) Use permutations and combinations to solve certain counting problems. 3) Compute probabilities using combinations.

3. SAT problems  In how many ways can you pick….

4. Counting Rule An easy way to find total possibilities or the SAMPLE SPACE

5. Counting Rule MULTIPLY

6. LICENSE PLATES  In Delaware, license plates have seven characters. How many possibilities are there total? What if you could only use numbers? What if you could only use numbers? What if you could only use letters? What if you could only use letters? What if you can’t repeat letters? What if you can’t repeat letters? What if you do not have any restrictions at all? What if you do not have any restrictions at all? 10·10·10·10·10·10·10= 10,000,000 26·26·26·26·26·26·26= 8,031,810,176 26·25·24·23·22·21·20= 3,315,312,000 36·36·36·36·36·36·36= 7.836 E 10

7. PASSWORDS  Most internet sites force you to have a password.  If one site requires 8 characters, letters only, how many possibilities are there? 26·26·26·26·26·26·26·26= 2.088 E 11

8. CELL PHONES  How many phone numbers are possible in Delaware?  Are there enough possibilities if everyone had their own cell phone? (There are about 850,000 people living in Delaware) 1·1·1·8·10·10·10·10·10·10= 8,000,000 (302)…-….

9. PERMUTATIONS Any arrangement of distinct objects in a particular order.

10. Factorial Notation 1) 3! = 3 x 2 x 1 2) 10! = 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 ON YOUR CALCULATOR MATH – PRB – !

12. Check this out!  PERMUTATIONS THE MOVIE PERMUTATIONS THE MOVIE PERMUTATIONS THE MOVIE

13. PERMUTATION FORMULA n = total r = how many you are selecting

14. Rankings  In how many different ways can you arrange the following beautiful ladies?

16. Sports  In how many ways can teams finish in the NFC East?

17. Starting Line UP  In how many ways can Coach Dinardo choose his starting offense from a total of 32 players?

18. Combination An arrangement of objects where order is NOT important.

19. Combination Formula  The number of combinations of n total things taken r at a time is denoted as

21.  Angel always dresses well. He is deciding which 2 shirts to wear to school this week out of his 13 clean shirts.  In how many ways can Angel choose which two shirts to wear?

22. SSSSerpe is cleaning his room and actually putting his collection of baseball cards away. He usually puts 4 new cards on display each month. IIIIn how many ways can he choose the 4 cards from a possible 32 cards?

23. Permutation vs. Combination  Both questions ask “in how many ways…”  Permutations need ORDER Ranking Ranking  Combinations DON’T More random More random

24. 1) The senior class is going to take the top four students (by GPA) to Jamaica for a little R&R. In how many ways can they be chosen? 2) What if they pick the four students randomly from a hat. In how many ways can they be chosen?

25. Poker!  How many different poker hands consisting of five cards can be dealt from a deck of 52 cards?  What is the probability of being dealt a royal flush in five card poker?

26. PASCAL’s TRIANGLE  It is a mathematical relationship between numbers that can be used for computing the number of possible combinations of n things taken r at a time.  Use Pascal’s Triangle to find the following.

27. 1 615 20 6 1