Bellwork Maria has an unidentified disease. She has the option to choose from three states in which to be treated. In each state, there are two research hospitals, and each hospital has three specialists that can help to diagnose her. How many different options does Maria have for healthcare? Draw a tree diagram to find the solution to the problem.

Section 7-1: Permutations and Combinations

Fundamental Counting Principle Definition: If one event can occur in m ways and a second event can occur in n ways, then the number of ways both events can occur is m·n Can this definition be extended to three events? Four events? More?

Examples Lunch Special “Choose 1 of each” Entrée Fried chicken Hamburger Chicken sandwich Side Dish Coleslaw Salad Fries Drink Milk Juice Water Soda How many different meal choices are there?

Most license plates consist of six number and letters. How many different license plates are possible if the six numbers and letters can be anywhere on the license plate? How many would be possible if the license plate number must start with three letters and end with three numbers? What if no letter or number can be repeated?

Permutation Definition: A selection of a group of objects in which order IS important. An ordering of n objects Permutations of n objects: n! = n· (n-1) ·(n-2) ·(n-3) ·... ·(3) ·(2) ·(1) The quantity 0! is defined as 1

Examples A school’s student council is electing new officers. From the total 70 members, only 8 are eligible to run for office. How many different ways are there to fill positions for president, vice president, secretary, and treasurer with eligible members?

How many different phone numbers are possible for the (402) area code? What if there are no restrictions on the area code for a ten-digit phone number?

How many ways can you select 3 people from a group of 7 people? 7 · 6 · 5 = 210 permutations We can also use factorials to find this answer: total arrangements = arrangements of 7 = 7! = 210 arrangements not used arrangements of 4 4! General Formula: n P r = n! = (n – r)!

A bride-to-be has 11 best friends. In how many ways can she select 5 of her best friends to be her bridesmaids? (Assume that the role of maid of honor has already been filled).

What happens when not all the objects are distinct? (in other words, some objects repeat). In general, the number of permutations of n objects where one is repeated q1 times, the second is repeated q2 times, and so on, is n! q 1 ·q 2 ·q 3 ·…·q n

In how many ways can you arrange the letters in OHIO? In MISSISSIPPI?

Combinations Definition: A grouping of items in which does not matter Generally, combinations are lesser quantities than their permutation counterparts? total arrangements = n C r = n! (arr. of selected)(arr. of unselected) r! (n-r)!

Examples Michael is going to California on vacation. His vacation package offers eight choices of fun activities to do each day he is there. If he has three days to do the activities, how many different ways can he choose his adventures?

At a track meet, there are five heats of the one mile race. The Creighton team has nine men racing. Three runners will be chosen to race in the first heat. How many ways can three runners be chosen from the team?

Guided Practice John is playing Yahtzee and he rolls his five dice. How many ways can he arrange the dice from left to right? How many ways can he choose 3 of the dice to reroll? Betsy is playing poker with her friends. How many different 5-card hands can she be dealt? How many different 5-card hands are possible, if all five cards must be of the same suit? Fred loves Qdoba, and he wants to order a burrito for dinner. Considering the following choices, how many different burritos could Fred make? Chicken, pork, shredded beef, steak, ground beef, vegetarian (pick one) Rice Black or pinto beans (pick one) Sour cream, cheese, guacamole, queso, grilled veggies, fajita veggies, lettuce Salsa roja, salsa verde, corn salsa, pico de gallo, habanero, mango salsa Chips or not

Textbook pg #2-14 even, #24-25

Talk with your neighbor What is the difference between a permutation and a combination? Give an example of when you would use each method.