1. Other Topping Sauce Ice Cream Vanilla Choc. Straw
Example: Metzger’s ice cream shop sells sundaes for $ Each Sundae consists of one flavor of ice cream, one sauce topping and one “other” topping. There are 3 flavors of ice cream, 3 sauce toppings, and 2 “other” toppings to choose from. How many possible sundaes can be made?
Other Topping
Sauce
Ice Cream
Vanilla
Choc.
Straw
Whipped Cream
Peanuts
Hot Fudge
Caramel
Butterscotch
6 with vanilla ice cream
Whipped Cream
Peanuts
Whipped Cream
Peanuts
6 with choc. ice cream
6 with straw. ice cream
Total of 18 types of sundaes.

2. Counting Principal: Number of ways that two or more events (ice cream, sauce, topping – 3 events) can happen
Multiply the number of ways each event can occur
Ice cream ex. (3)(3)(2) = 18

3. Ex. Ashley bought 4 shirts, 3 pairs of pants, 2 pair of shoes, and 3 pair of earrings on her shopping spree. How many outfits can she create using her “new” items?
Name the events:
shirts, pants, shoes, earrings
(4)(3)(2)(3) = 72 different outfits.

4. Ex. A combination lock has 40 numbers on its dial
Ex. A combination lock has 40 numbers on its dial. The lock has a 3 number combination. How many combinations are possible?
Name the events:
1st # in comb., 2nd # in comb., 3rd number in comb.
How many possibilities for each event?
1st # = 40 , 2nd # = 40 , 3rd # = 40
How many possible combinations:
(40)(40)(40) = 64,000 possible combinations
You randomly choose 3 numbers for the combination, what is the probability that you choose the correct combination on the first try?

5. Example : How many ways can 3 people stand in a line at Walmart?
Lesson 11.6 : Permutations
ORDER MATTERS
Permutation: arrangement of objects where order MATTERS
(n objects using r of them at a time)
Example : How many ways can 3 people stand in a line at Walmart?
3 objects, using 3 at a time
(#can be 1st)(#can be 2nd after one chosen for 1s)(#can be 3rd after
chosen for 1st and 2nd)
6 ways

6. Example: You have 4 posters to hang in your room
Example: You have 4 posters to hang in your room. You want to put one poster on each wall. How many ways can you arrange the posters
4 objects using 4 at a time.
W W W W 4
4 choices• 3 choices • 2 choices • 1 choice
24 ways to hang posters

7. Example: You have 7 posters to hang in your room
Example: You have 7 posters to hang in your room. You want to put one poster on each wall. How many ways can you arrange the posters
7 objects using 4 at a time.
P P P P 4
7 choices • 6 choices • 5 choices • 4 choices
840 arrangements

8. Ex. There are 8 swimmers in the 200 meter race
Ex. There are 8 swimmers in the 200 meter race. First place receives $100, second place receives $50 and third place receives $25. How many different ways can the first, second, and third place prizes be awarded?
Number of items = 8
Number taken at a time = 3
(8)(7)(6) = 336 ways

9. What if ORDER DOESN’T MATTER?
Example: You want to buy 4 different CD’s. You can afford to buy only 3 of them. How many combinations of CD’s can you buy?
1, 2, 3
1, 2, 4
1, 3, 4
2, 3, 4
4 possibilities – eliminate duplicates like 1, 2 ,3 and 2, 1, 3

10. CD ex. r! = means “r factorial”
Combination: Order does NOT matter – get rid of repeats
Combination of n objects taking r at a time
r! = means “r factorial”
Factorial: 5! = 5  4  3  2  1 = 120
CD ex.

11. Ex. Papa Johns offers 12 different toppings, how many 3-topping pizzas are possible?
= 220
There are 220 possible 3-topping pizzas.

12. Ex. There are 6 essay questions on a test. You must choose 4 to answer
Ex. There are 6 essay questions on a test. You must choose 4 to answer. How many different groups of 4 questions could be chosen?
= 15
There are 15 different groups of 4 questions.

13. Ex. Jim rented 8 movies to watch over the weekend
Ex. Jim rented 8 movies to watch over the weekend. He only has time to watch 5 of the movies. How many ways could he choose 5 movies to watch?
= 56
There are 56 possibilities.

14. To determine if Permutation or Combination:
Ask yourself: DOES ORDER MATTER?
Yes = Permutation (no dividing)
No =Combination (need to divide to get rid of repeats)
Homework : worksheet