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Explore the intriguing world of combinations and permutations, focusing on scenarios where order matters. From calculating the total possible passcodes for locks with specific digit limitations to understanding license plate configurations, this guide examines essential principles in combinatorial mathematics. Discover how to model different situations such as phone numbers, club officer selections, and even lunch orders, highlighting when order is key and when it isn't. Enhance your understanding of these mathematical concepts in practical applications.
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Possibilities Order - when does it matter? Combinations and Permutations
Locks??? Consider… Does order matter?
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 ___ ___ ___ ___
License Plates??? Consider… Does order matter?
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?
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
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
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!
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
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:
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
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
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?
