Permutations and Combinations

Random Things to Know Dice (singular = “die”) Most cases: 6 sided Numbers 1,2,3,4,5,6 𝑃 𝐴 = 1 6 Special Cases: 4 sided 8 sided 10 sided 12 sided 20 sided

Random Things to Know Cards Typical Deck: 52 cards 4 Suits (13 cards each) Clubs Spades 2-10 2-10 3 Face 3 Face 1 Ace 1 Ace Hearts Diamonds

Random Things to Know

Counting Principle If you have 6 shirts and 3 pants how many different outfits can you create?

Counting Principle If you have M of one option and you have N of another option, Then there are 𝑴∙𝑵 ways of doing both M = shirts N = pants Number of outfits you can make = 𝑀∙𝑁 6∙3=18

Counting Principle When flipping a coin 15 times how many results are possible? *Think how many different results are there when you flip a coin*

Counting Principle A restaurant has on its menu 5 choices for appetizers 3 choices for main course 2 choices for dessert How many different meals (appetizer, main course, and dessert) can you choose?

Factorials!  

Let’s Practice…… A student is to roll a die and flip a coin. How many possible outcomes will there be? For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from?

Permutations You and your 3 friends are standing in line to buy tickets to a movie. How many ways are there for you to arrange yourselves?

Permutations Remember the Counting Principle: M*N = total number of ways to select items How many choices do you have for the first spot? (4) How many choices do you have for the second spot? (3) How many choices do you have for the third spot? (2) How many choices do you have for the fourth spot? (1) So 4∙3∙2∙1=24

Permutations Def: A way of selecting items where the order does matter In races who comes in 1st, 2nd, and 3rd is very important for prizes, and rankings. The order does matter.

Permutations

Permutations You can use your calculator to find permutations To find the number of permutations of 10 items taken 6 at a time (10P6): Type the total number of items Go to the MATH menu and arrow over to PRB Choose option 2: nPr Type the number of items you want to order

Let’s Practice…… Find the number of ways to arrange the letters ABC. A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

BIG QUESTION: DOES ORDER MATTER?? Combinations If you have 5 trophies but only space on a shelf for 2 of them how many different ways can you arrange your trophies? BIG QUESTION: DOES ORDER MATTER??

Combinations Remember the Counting Principle: M*N = total number of ways to select items How many trophies can you choose between? 5 How many spots are there? 2 So… 5∙2=10

Combinations Def: A way of selecting items where the order does not matter If you order pizza it doesn’t matter if you tell them “Peperoni, Pineapple, and Sausage” or “Sausage, Peperoni, and Pineapple” NO! It all goes on the pizza! The order doesn’t matter

Combinations 𝑛 ∁ 𝑟 𝑜𝑟 𝑛 𝑟 = 𝑛! 𝑟! 𝑛−𝑟 ! n = total number of elements 𝑛 ∁ 𝑟 𝑜𝑟 𝑛 𝑟 = 𝑛! 𝑟! 𝑛−𝑟 ! n = total number of elements r = number of items chosen 𝑛 ∁ 𝑟 = 5! 2! 5−2 ! = 120 2(6) =10

Combinations You can use your calculator to find combinations To find the number of combinations of 10 items taken 6 at a time (10C6): Type the total number of items Go to the MATH menu and arrow over to PRB Choose option 3: nCr Type the number of items you want to order

Let’s Practice…… To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible? A basketball team consists of two centers, five forwards, and four guards. In how many ways can the coach select a starting line up of one center, two forwards, and two guards?