. is order important?. is order important?

. how do you compute these?. how do you compute these?

PermutationsPermutations

Let’s look at an example!Let’s look at an example!At a juice bar, a customer who orders a smoothie gets to choose one type of fruit At a juice bar, a customer who orders a smoothie gets to choose one type of fruit

and one type of yogurt. For example, customers who order a “Simple and one type of yogurt. For example, customers who order a “Simple Smoothie” get to choose from 3 fruits (bananas, peaches, and blueberries) and Smoothie” get to choose from 3 fruits (bananas, peaches, and blueberries) and 2 yogurts (plain or vanilla). The tree diagram shows that there are 6 possible 2 yogurts (plain or vanilla). The tree diagram shows that there are 6 possible types of Simple Smoothies:types of Simple Smoothies:

plainplain plainplain plainplain

Banana Banana Peach Peach Blueberry Blueberry

vanillavanilla vanillavanilla vanillavanilla

A permutation is a selection of items from a group in which the order is important. In a permutation, AB is not the same as BA.

To begin, consider there are n items and you want to select r items from that group and organize them in a certain order. The mathematical formula is You may also think of this as (n)(n-1)(n-2)… multiplied r times.

For instance, if there are 9 items and you want to order 3 of them, you can multiply 9x8x7 and this would give you the number of ways to order the 3 items (the factorial formula also gives the same result).

Want to give this a try on your own?(click when you’re ready…)

How many ways can club members select a President, a Vice President, a Secretary, and a Treasurer from a group of 10 members? Remember, order matters because each office is different…

)!(

!Pr

rn

nn

type your answer in the Blackboard free response area for your teacher to see…

(I’ll be waiting…)

What’s your guess?

If you answered 50405040, you were correct!

If you didn’t get that answer, no worries mate!

We’ll try another one now…

““Those Pesky Passwords”Those Pesky Passwords”

Nowadays it seems you need a password to sign into every Nowadays it seems you need a password to sign into every internet site. Mrs. Herr’s math website is no different; she internet site. Mrs. Herr’s math website is no different; she requires a password of 5 letters, with only the capital letters A requires a password of 5 letters, with only the capital letters A thru H being valid. Letters may thru H being valid. Letters may notnot be repeated be repeated. .

How many possible passwords are there to her website?How many possible passwords are there to her website?

Some valid examples are DECAH, BEACH, and FACED..

Invalid entries would be DEADF (repeating D’s), HEADS (S Invalid entries would be DEADF (repeating D’s), HEADS (S isn’t between A and H), and ABcDe (lower case letters not isn’t between A and H), and ABcDe (lower case letters not permitted)permitted)

Let’s get to the answer… There are 8 choices of letters – A, B, C, D, E, F, G, and H There are 5 positions to be filled for the password _ _ _ _ _ You have 8 choices of letters for the 1st position… 7 letters to pick for the 2nd position… 6 choices for the 3rd position, etc… So there are 8x7x6x5x4 or 6720 possible passwords

Did you get that for your answer?

Way to go, buddy! I knew you could do it!Way to go, buddy! I knew you could do it!

Next, we’re off to discoverNext, we’re off to discover

COMBINATIONSCOMBINATIONS

CombinationsCombinations

A combination is a selection of items from a group in which order is not important. AB is the same as BA; ‘xyz’ is the same as ‘xzy’, ‘yxz’, ‘yzx’, ‘zxy’, and ‘zyx’.

Here’s what a combination problem Here’s what a combination problem looks like:looks like:

How many ways How many ways can club members can club members select a committee select a committee of 4 people from a of 4 people from a group containing group containing 10 members?10 members?

The order of the members does not matter… selecting Tom, Joe, Sally and Molly is the same as selecting Sally, Joe, Molly and Tom.

Therefore a formula different from the permutation formula must be used. It is similar but has one key difference:

(notice the r! in the denominator ~ that’s not present in the permutation formula).

)!(!

!

rnr

nnCr

Melania is married to a wealthy man who Melania is married to a wealthy man who has showered her with many expensive has showered her with many expensive items. Among them is a collection of 20 items. Among them is a collection of 20 precious-jewel rings. precious-jewel rings. Melania wants to wear 3 of these but just Melania wants to wear 3 of these but just can’t decide which ones to select. How can’t decide which ones to select. How many groups of 3 rings does she have to many groups of 3 rings does she have to choose from?choose from?

<enter your answer in the response area of Blackboard>

your answer here:

In all of the combinations of 3 rings, order is not important…picking the sapphire, diamond, and ruby rings is the same as picking the ruby, diamond and sapphire rings. So the answer is obtained by using the combination formula:

How’d you do? AWESOME!!

1140!17!3

!20

)!(!

!

rnr

n

Thanks for your time – peace out!Thanks for your time – peace out!