Possibilities

1. Possibilities Order - when does it matter? Combinations and Permutations

2. Locks??? Consider… Does order matter?

3. Possible # of passcodes? # of possible passcodes Possibilities for each digit Restrictions • 5 digits • #’s 0 – 7 for each • #’s can repeat 8 8 8 8 8 32,768 ___ ___ ___ ___ ___ 39 39 • 3 digits • #’s 0 – 39 • Consecutive #’s cannot repeat 40 ___ ___ ___ 60,840 • 4 digits • 1st – Letters “I to P” • 2nd – Letters “Q to X” • 3rd – Letters “A to H” • 4th - #’s 0 to 7 4096 8 8 8 8 ___ ___ ___ ___

4. License Plates??? Consider… Does order matter?

5. License Plates??? California Restriction example… • 1st - #’s 1 to 7 • 2nd, 3rd, & 4th – Letters from “A to Z” • 5th, 6th, & 7th - #’s from 0 to 9 10 10 26 7 26 10 26 ___ ___ ___ ___ ___ ___ ___ Total possibilities = 123,032,000 Does order matter?

6. What about??? A 4 digit passcode with the following requirements… • 1st digit must be a capital letter. • 2nddigit can be either a lowercase or capital letter. • 3rd digit must be a number from 0 to 9. • Last digit must be one of the following special symbols: ~, !, @, #, $, %, &, ? 8 ____ ____ ____ ____ 52 26 10 Total Possibilities = 108,160

7. What about??? President, Vice President, and Treasurer for a club taken from a group of 11 people… 10 11 ____ ____ ____ Pres VP Treas 9 Total Possibilities = 990 Phone numbers beginning with the same area code, disregarding any special restrictions such as starting with a 1 or 0, 911, 411, operator, etc. ____ ____ ____ - ____ ____ ____ ____ 10 10 10 10 10 10 10 Total Possibilities = 10,000,000

8. What if problem is too difficult to model?Or, you prefer to use a formula? Order matters – called a Permutation • Ex: 5 letters arranged in groups of 3, or 5 items taken 3 at a time • ABC and CBA would be considered 2 different possibilities • Formula: NOTICE – answer is the 1st k factors ofn!

9. Permutations… • 10 out of 12 individuals seated in a row. • First 5 students called from a group of 50. • The letters of the word “Sabres” arranged to form different passwords. 239,500,800 254,251,200 720

10. What if problem is too difficult to model?Or, you prefer to use a formula? Order doesn’t matter – called a Combination • Ex: 5 letters arranged in groups of 3, or 5 items taken 3 at a time • ABC and CBA would be considered the same possibility. • Formula:

11. Based on this how many possibilites for each: • Pizza w/ choice of 5 toppings (pick 2) • Pizza w/ choice of 8 toppings (pick 5) 10 56

12. Ordering lunch… does order matter? • Sandwich: Choice of wheat, white, or rye bread with choice of Ham or Turkey. • Side: Choice of baked or regular chips. • Drink: Choice of Coke, Sprite, Dr. Pepper, or Tea. Hmmm… does the order of my request affect my actual lunch? I could get the total lunch possibilities by… _____ _____ _____ _____ Bread Meat Chips Drink 3 2 2 4 48 Total Lunch Possibilities

13. Ordering lunch… does order matter? This problem is a Combination problem… however, with each choice you are only selecting 1 item. When the case, permutation or combination yields same result. _____ _____ _____ _____ Bread Meat Chips Drink 3 2 2 4 48 Total Lunch Possibilities Which is easier method when only choosing one of each choice?