Warm Up – 4/22 - Tuesday A deli has a lunch special which consists of a sandwich, Soup, dessert, and a drink. They offer the following choices: Sandwich: chicken salad, ham, and tuna, and roast beef Soup:  tomato, chicken noodle, vegetable Dessert: cookie and pie Drink: tea, coffee, coke, diet coke and sprite Each lunch special costs $4.99. How many different lunch specials could one order? How much would ordering every lunch special cost?

Fundamental Counting Principal More simply, the total number of choices is the product of your number of individual choices.

Example #1 Suppose Mr. Gill is going to rearrange the seating chart in the classroom. He has four empty desks for Gabe, Matt, Katie, and Nicole. How many different ways can he arrange them?

Factorials What we will see often is scenarios that do not have 𝑟𝑒𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 meaning once I have obtained the result once, I cannot get the result again. In our last example, suppose Mr. Gill puts Matt in the front row. Matt is no longer an option meaning my number of choices has decreased from four to three.

Factorials Every time I place someone I lose one of my options until I get to zero options left. At the beginning: 4 options Place 1 person: 3 options Place 2nd person: 2 options Place 3rd person: 1 option Using the FCP (Fundamental Counting Principal), I can see that my total options are: 4 𝑥 3 𝑥 2 𝑥 1 OR: 4!

Example #2 Suppose Mr. Gill is arranging his math book collection on the book shelf. If Mr. Gill owns 8 math books, how many ways can he arrange them on the shelf?

Operations with Factorials What is a simpler way for me to rewrite the following? 7∙6∙5! 5! 3! 10! 7−2 !

Fundamental Counting Principal Worksheet

Permutations

Example #1 - Repetition Allowed Billy is late for class and is trying to break into his friend’s bike lock. The bike lock has the digits 0-9. How many possible sequences are there if the lock can use the same number more than once? (Ex: 333)

Repetition Allowed We can say there are 10 possibilities for each column. Using the FCP: 10 𝑥 10 𝑥 10=1000 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠

Example #2 - Repetition NOT Allowed Let’s change the problem so that now we cannot use the same number twice. How many possible sequences are there?

Repetition NOT Allowed In this case we have 10 choices, then 9 choices, then 8 choices. Using the FCP: 10 𝑥 9 𝑥 8=720 𝑠𝑒𝑞𝑢𝑒𝑛𝑐𝑒𝑠 What does 10 𝑥 9 𝑥 8 look like?

Permutations Formula In our last example, we had 10 digits to choose from, and we needed 3 digits. 𝑛=10, 𝑟=3.

Example #3 Suppose 10 people are running a race. The racing council awards a Gold, Silver, and Bronze medal to the top three finishers. How many different ways could we award these medals?

Permutations Homework

The probability of simple events

Example

Solutions As you can see for the spinner, there are 6 possible outcomes. This will always be our denominator. 1. 𝑃 𝐶 = 1 6 =0.166=16.7% 2. 𝑃 𝐺 = 0 6 =0=0% 3. 𝑃 𝑀 𝑜𝑟 𝑃 = 2 6 =.333=33.3% 4. 𝑃 𝐵, 𝐸, 𝑜𝑟 𝐴 = 3 6 =.50=50%