Lesson 7-1 Permutations & Combinations

Lesson 7-1 Permutations & Combinations

Poll

What is this?

• Index

• Tree Diagram

• Table of Values

Tree Diagram

A tree diagram gives a picture for various groupings

Fundamental Counting Principle

If there are n items and m1 ways to choose a first item, m2 ways to choose a second item after the first item has been chosen, and so on, then there are m1 x m2 x .... mn ways to choose n items

Poll

To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there?

• 10

• 20

• 30

• 60

Multiple Choice

A "make-your-own-adventure" story lets you choose from 6 starting points, gives 4 plot choices, and then has 5 possible endings. How many adventures are there?

• 24

• 48

• 60

• 120

Multiple Choice

A password for a site consists of 4 digits followed by 2 letters. The letters A and Z are not used, and each digit or letter may be used more than once. How many unique passwords are possible?

• 3,260,000

• 5,760,000

• 6,460,000

• 8,560,000

Open Ended

Write down an example of when 'order' matters. For example, when we looked at composite functions, the order mattered. Another example of when order matters is for the password on your phone.

! Factorial

• See that the number of permutations of 3 items is

   $3\ \cdot\ 2\ \cdot\ 1$  

• Extend this to  $n\ \cdot\ \left(n-1\right)\ \cdot\ \left(n-2\right)\ \cdot...\ \cdot1$  

• This is called  $n\ factorial$  and is written as n!

Permutations

• We can use factorials to find possible permutations, which leads to a formula

• Useful for large numbers

• Divide total number of arrangements by the number of arrangements not used

Poll

How many ways can a stylist arrange 5 of 8 vases from left to right in a store display? [Before solving this, how do we set up the equation?]

• $\frac{8!}{\left(5!\right)}$

• $\frac{8!}{\left(8-5\right)!}$

 $\frac{8!}{\left(8-5\right)!}$  

How many ways can a stylist arrange 5 of 8 vases from left to right in a store display?

The total number is 8, so use 8! for the numerator.

The amount not used is (8-5), so use (8-5)! for the denominator.

Poll

How many ways can a stylist arrange 5 of 8 vases from left to right in a store display?

• 336

• 6720

• 40,320

6720

 $\frac{8!}{3!}=6720$  

Poll

Awards are given out at a costume party. How many ways can "most creative," "silliest," and "best" costume be awarded to 8 contestants if no one gets more than 1 award?

• 336

• 6720

• 40320

336

 $\frac{8!}{\left(8-3\right)!}=336$  

Multiple Choice

How many ways can a 2-digit number be formed by using only the digits 5 - 9 and by each digit being used only once?

• 10

• 20

• 60

• 120

Solving a Combination

• Simplifying Fractions is a Great Skill

• Use ! on Calculator

Poll

The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected? (Before you answer the question, you must decide if this is a permutation or a combination?)

• Permutation

• Combination

Combination

Order does not matter for the swimmers

Multiple Choice

The swim team has 8 swimmers. Two swimmers will be selected to swim in the first heat. How many ways can the swimmers be selected?

• 16

• 28

• 56

• 20,160