INTRO TO COUNTING: 11.4 Day 1

SIMPLE ENUMERATION COLLECTION OF ITEMS THAT IS A COMPLETE, ORDERED LISTING OF ALL OF THE ITEMS IN THAT COLLECTION. How many squares on a checkerboard? You could count…. or….. 8x8 = 64

COMBINED OUTCOMES Suppose we flip a coin and then throw a single die. What are the possible combined outcomes? How can we organize our counting?

COMBINED OUTCOMES Suppose we flip a coin and then throw a single die. What are the possible combined outcomes?

AN EASIER APPROACH Suppose we flip a coin and then throw a single die. What are the possible combined outcomes?

MATH SELFIES You are at a math party and would like to take a bunch of trig selfies to post on Facebook. How many selfies will you need to take if you wish to have a selfie that includes you and one each of: 4 friends, 3 backgrounds, and 6 math books?

MULTIPLICATION PRINCIPLE OF COUNTING If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the total number of selections possible is determined by:

A BIT HARDER… From the 26 letters in the alphabet, how many ways can 4 letters appear in a row on a license plate if no letter is repeated?

A BIT HARDER… From the 26 letters in the alphabet, how many ways can 4 letters appear in a row on a license plate if no letter is repeated?

A BIT HARDER… From the 26 letters in the alphabet, how many ways can 4 letters appear in a row on a license plate if no letter is repeated?

LET’S CHANGE THIS QUESTION JUST A LITTLE Ex. 2) From the 26 letters in the alphabet, how many ways can 4 letters appear in a row on a license plate if we ALLOW the letters to repeat?

LET’S CHANGE THIS QUESTION JUST A LITTLE From the 26 letters in the alphabet, how many ways can 4 letters appear in a row on a license plate if we ALLOW the letters to repeat?

YOU TRY 7-digits such that the first and last digits may be any integer 1 through 9, and the middle five digits may be any integer 0 through 9. How many of these plates are possible? A letter (A, J, X, B, C, D, F, T, Y, G, H, K, L, N, P, R, S, or V) followed by 6-digits such that each digit may be any integer 0 through 9. How many of these plates are possible? Licensesnon-personalized Illinois license plates come in two forms:

Dependent Events: The first event affects the outcome of the next event and the next event needs to be adjusted accordingly. Ex) Each player in Monopoly uses one of 6 pieces. How many ways can four people choose a board game piece? Independent Events: The first event does not affect the outcome of the next event Ex) Mary had 5 concert t-shirts and 6 pairs of pants to wear to a party. How many ways can she make an outfit from these clothes INDEPENDENT VS. DEPENDENT EVENTS

Dependent Events: The first event affects the outcome of the next event and the next event needs to be adjusted accordingly. Ex4) Each player in Monopoly uses one of 6 pieces. How many ways can four people choose a board game piece? Independent Events: The first event does not affect the outcome of the next event Ex3) Mary had 5 concert t-shirts and 6 pairs of pants to wear to a party. How many ways can she make an outfit from these clothes INDEPENDENT VS. DEPENDENT EVENTS

Area Codes Initial use of three-digit area codes in the United States and Canada began in 1947 in large cities. Without any restrictions, how many three-digit area codes are possible?

Area Codes Initial use of three-digit area codes in the United States and Canada began in 1947 in large cities. Without any restrictions, how many three-digit area codes are possible?

A compound event is one event that has more than one simple event. Mary may choose one snack from 3 different pieces of fruit or 2 different types of granola bars. **Mary is only picking one thing** The Adding Principle & Compound Events

Mutually Exclusive Events Two Events CANNOT happen at the same time Mutually Inclusive Events Two Events CAN happen at the same time

Mutually Exclusive Events Two Events CANNOT happen at the same time Examples: 1. How many ways can you roll and even OR an odd? 2. How many ways can you draw one card from a 52-card deck and get an 8 OR a King? You add together the number of ways for each simple event

Mutually Inclusive Events Two Events CAN happen at the same time Examples: 1. How many ways can you roll an odd OR a multiple of 3 on a die. 2. How many ways can you draw an 8 or a Heart from a 52-card deck 3. There are 28 kids in a class. 15 are freshman and 13 are sophomore. 10 are boys and 20 are girls. 7 of the boys are sophomores. How many ways can a teacher call on a sophomore or a boy. This occurrence of both events is called the OVERLAP.

Answer: M = Number of ways for first event to occur N = Number of ways for second event to occur M ∩ N = Number of ways they occur together, or the overlap. # of ways = M + N – (M ∩ N) – 1 = 16 ways ** How many ways can you draw an 8 or a Heart from a 52-card deck?

Add or Multiply?? Example #8 Penelope is ordering a pizza. She may choose one from each of the following: 3 types of crusts, 4 types of cheeses, and 6 types of vegetables. How many possible pizzas could she create? Example #9 Barney is choosing a book to read at the library. On the shelf there are 3 fiction books, 5 autobiographies, or 2 history books. How many ways are there for him to choose a book?

Add or Multiply?? Example #1 Penelope is ordering a pizza. She may choose one from each of the following: 3 types of crusts, 4 types of cheeses, and 6 types of vegetables. How many possible pizzas could she create? Example #2 Barney is choosing a book to read at the library. On the shelf there are 3 fiction books, 5 autobiographies, and 2 history books. How many ways are there for him to choose a book?

Add or Multiply?? Example #10 The senior class is voting for President and Vice- President out of 10 total candidates. How many ways are there to choose a President and then a Vice-President? Example #11 A scholarship committee is selecting a student from District 86 to win $1000 for college. There are 25 eligible students from Hinsdale Central and 29 eligible students from Hinsdale South. How many ways are there to award this scholarship?

Add or Multiply?? Example #3 The senior class is voting for President and Vice- President out of 10 total candidates. How many ways are there to choose a President and then a Vice-President? Example #4 A scholarship committee is selecting a student from District 86 to win $1000 for college. There are 25 eligible students from Hinsdale Central and 29 eligible students from Hinsdale South. How many ways are there to award this scholarship?

Challenge! At first, area codes were all in the form NYX, where N is any integer 2 through 9, Y is 0 or 1, and X is any integer 1 through 9 (if Y is 0) or any integer 2 through 9 (if Y is 1). (In other words, 312 was okay, but 311 was not.) With these restrictions, how many area codes were possible? More with area codes…

Challenge! At first, area codes were all in the form NYX, where N is any integer 2 through 9, Y is 0 or 1, and X is any integer 1 through 9 (if Y is 0) or any integer 2 through 9 (if Y is 1). (In other words, 312 was okay, but 311 was not.) With these restrictions, how many area codes were possible? Answer: Why the restrictions? The restriction on N saves 0 for calling the operator, and 1 for signaling a long-distance call. The restriction on the second digit, limiting it to 0 or 1, was designed to help telephone equipment recognize the difference between a three-digit "area code" (with 0 or 1 as the second digit) and the three-digit "exchange" prefix (which had avoided 0 or 1 for the second digit, because of restrictions in existing switching equipment). calling the operator

BRAIN BREAK

Students have to move their right foot in a clockwise circle, and then with their right pointer finger, they need to write the number 6 in the air. GO SLOW…repeat with left foot and left pointer.

Phone Numbers 12. How many 7 digit phone numbers are possible if the number cannot start with zero? 13. How many possible area codes exist if you must exclude 911? License Plates 14. Illinois license plates can have 7 numbers. How many license plates are possible? You may repeat numbers 15. Michigan license plates can have 3 numbers and 3 letters. How many of these plates are possible?

Phone Numbers 1)How many 7 digit phone numbers are possible if the number cannot start with zero? 2) How many possible area codes exist if you must exclude 911? License Plates 1)Illinois license plates can have 7 numbers. How many license plates are possible? You may repeat numbers 2) Michigan license plates can have 3 numbers and 3 letters. How many of these plates are possible?

Factorials! What does 4! mean? Find 11! With your calc These can be a more efficient way to apply the multiplication principle. Ex) How many ways are there to arrange 13 kids in 13 different chairs?

Factorials! What does 4! mean? Find 11! With your calc These can be a more efficient way to apply the multiplication principle. Ex) How many ways are there to arrange 13 kids in 13 different chairs?

Understanding why: 0! = 1 Suppose you want to line up: 1 items Suppose you want to line up: 2 items Suppose you want to line up: 0 items Suppose you want to line up: 3 items

Understanding why: 0! = 1 Suppose you want to line up: 1 items Suppose you want to line up: 2 items Suppose you want to line up: 0 items Suppose you want to line up: 3 items

Manipulating Factorials!

Challenge!!

Can you simplify:

Challenge!! Can you simplify:

Challenge!! Can you simplify:

Challenge!! Can you simplify: