Permutations

Anagrams

SCRABBLEGRAMS
& ANAGRAMS

Anagrams was a popular game 100 hundred years ago. You pick a word and rearrange its letters in as many ways as you can. Scrabble evolved from anagrams. Today, we can play ScrabbleGrams everyday in the daily paper.

How many ways can you arrange 7 different Scrabble tiles?

Fill Out This Table:
Show how you would count the number of ways
to arrange 1, 2, 3, 4, 5, 6, and then 7 Scrabble tiles.
Do you have a calculator powerful enough to count the arrangements of 15 different letters?

The numbers you listed are called permutations.
A permutation is a distinct order of a set of things.
A permutation is another word for arrangement.
ABC, CAB, and BAC are each permutations of the letters ABC.
According to the Fundamental Principle of Counting,
if we have 3 things, we can arrange them, in 3 x 2 x 1 = 6 ways.
If we have n different things, we can arrange them in n x n-1 x n-2 x ... x 1 ways.
Instead of writing n x n-1 x n-2 x ... x 1 ways, which is rather long,
we use the shorthand equivalent n!, read n factorial .
The number of ways to arrange the letters ABC are 3 x 2 x 1 = 3! or 3 factorial.

Click & Drag Letters

Most of the time, the dictionary contains no 7 letter word for our 7 Scrabble tiles.
Sometimes we do well to make a 3 or 4 letter word from 7 letters.

How many 3 letter arrangements can you make from 7 letters?

How many 4 letter arrangements can you make from 7 letters?

How many k letter arrangements can you make from 7 letters?

Fill Out The Table Below:
Let the numbers in the k column be the letters we take out of the 7 we are given.

Let the symbol _nP_k mean that we have n objects taken k at a time.

When k=3, the number of permutations we could make of 3 things selected from 7 things is
7 x 6 x 5=210.
Another way to say the same thing:
"The number of permutations of 7 things taken 3 at a time is 210."
Or using shorthand, ₇P₃ = 210.

IMPORTANT! When we count permutations, we make these assumptions:

All n objects are different.
None of the k objects is repeated.
Order of the k objects is counted.

QUESTIONS: PART 1

HOW MANY WAYS CAN YOU ...

Visit the capitals of each of the 50 states?
Shuffle a 52 card deck?
Place 3 prisoners in a triangle facing each other?
Seat 4 ladies in 4 chairs?
Visit 5 cities starting at A and returning home without driving any road twice?
Arrange the letters of the alphabet?
Arrange the digits in your phone number? Assume they are all different.
Sing the first 8 notes of "Joy to the World".
Arrange the batting order of 9 baseball players?
Arrange the 10 digits: (0 1 2 3 4 5 6 7 8 9)?
Place 11 football players in 11 positions?
Seat 12 players at a round poker table relative to each other?
Stack 13 boxes?
Shelve 14 books?
Queue 15 people in a ticket line?
Sequence 16 truth tables?
Hang 17 pictures?
Deliver 18 pizzas to 18 homes starting from the Pizza Hut?
Assign 19 different jobs to 19 workers?
Draw lines between 20 points so that each point is connected to exactly 2 other points?

QUESTIONS: PART 2

Assume that all n objects are different.
None of the k objects is repeated.
Order of the k objects is counted.

HOW MANY WAYS CAN YOU ...

Make no decisions and choose to do nothing?
Pick 1 letter from the alphabet?
Play 2 songs out of a collection of 52?
Place 3 X's on a tic-tac-toe grid?
Choose 4 linebackers from 11 football players for specific positions?
Deal 5 cards from a 52 card deck?
Select 6 infielders for 6 positions from 9 baseball players?
Visit 7 out of 10 cities on your vacation?
Shop 8 stores out of 10 in the mall?
Pick 9 players out of 20 for baseball positions?
Visit 10 state capitals?
Pick 11 players out of 20 for football positions?
Eat a dozen donuts from a box of 12?
Decide which of 13 homes to deliver 4 Pizza Hut pizzas?
Pick 4 books to read from a shelf containing 14 books?
Hire 4 people for 4 jobs from 15 applicants?
Elect a President, Vice-President, Secretary, and Treasurer from a fraternity with 16 members?
Watch 4 Blockbuster movies out of 17 new ones?

CLASS ASSIGNMENT

Pick 5 cities you want to visit any where in the world. Write them down.
How many ways can you visit them?
If you can only budget a visit to 3 of them,
how many ways can you choose to visit them?
Count the number of desks in our classroom.
Count the number of students in class.
How many possible ways can all the students be seated?
If there were 5 desks in the room and 3 students,
how many possible ways could the students be seated?
List all the possible ways in a table.
Draw a tree diagram (or a flower) showing all the permutations.

Hint: When starting to think about a counting problem, especially one with big numbers, always ask these questions based on the Fundamental Counting Principle:

How many ways can you choose the first object?
How many ways can you choose the second object?
How many ways can you choose the third object?
How many ways can you choose the fourth object?
.
.
.

WEB ASSIGNMENT

List all the different letters in your first name.
List all the different letters in your last name.
Select the list with the most letters.
How many letters are in your list?
How many ways can you arrange these letters? Show why.
How many ways can you pick 4 letters to arrange from these letters?
Show how you derived your answer.
Illustrate your answer with the Flower mathlet.
Email your instructor a screencopy of the results.
Each time you email a correction, include ALL the previous steps 1 to 9;
otherwise your instructor will not know which directions you have previously completed.

CHECK LIST
Your illustration must contain these properties:

The number of tips on your flower must equal the answer to question 6 above.
Each number in your STEM box must correspond to a number in your calculation, and vice versa.
Each number in your calculation must be represented by a different color in your diagram.
This means you must have 4 colors: one for each of the letters you choose.
Your colors must be different from the example.
Your radii lengths must be different from the example.
Include the input boxes with your drawing so we know what you did to draw your flower.
Screencopy and email your flower to your instructor for 5 points.
Include your FIRSTNAME and LASTNAME and the letters you used.
Answer the questions.
Show your calculations.
Send your illustration and answers in early and often so your instructor can correct you.
Re-read these directions. You probably overlooked something.
Don't lose points unnecessarily.