1. Counting Theory (Permutation and combination)
Section 6.0.1
Lesson 6.0.1

2. Starter 6.0.1
Suppose you have 6 different textbooks in your backpack that you want to put on a bookshelf. How many ways can the 6 books be arranged on the shelf?

3. Objectives
Use organized lists and tree (branching) diagrams to list all possible outcomes of a trial.
Identify whether permutation or combination is appropriate to count the number of outcomes of a trial.
Use formulas or calculator commands to evaluate permutation and combination problems.

4. Counting Outcomes
There are three principle ways to count all the outcomes of a trial.
Draw a tree diagram
Often a simple multiplication is enough
Systematically write all possibilities
Use permutation and combination techniques

5. Example: Tossing Coins
Three coins are tossed (or one coin is tossed three times) and the outcome of heads or tails is observed.
Draw a tree diagram (also called a branching diagram) that shows all possible outcomes.
State a conclusion: How many equally likely outcomes are there in this problem?

6. Tree Diagram for 3 Coins
First Toss
Second Toss
Third Toss H T H H H T T H H T T T H T
So there are 8 different equally likely outcomes.

7. Write an Organized List
For the coin-toss problem we just did, write an organized list that shows all possible outcomes (like HHH etc)
Here is one possible organization
HHH, HHT, HTH, HTT
TTT, THT, TTH, THH
There are other ways to organize
Any method that is systematic (so that no outcomes are missed) can work

8. Permutations of n objects
Return to the bookshelf question.
Suppose we change the problem to arranging 10 books on the shelf. Now how many arrangements are there?
10 x 9 x 8 x … = 3,628,800
The shorthand notation for this is 10! (factorial)
You can evaluate it quickly by MATH:PRB:4 (!)
In general, there are n! ways to arrange n objects
This is called the permutation of n objects
The key idea is that order matters

9. Arranging fewer than all the objects
What if there are only 4 slots available on the bookshelf for the 10 books?
Then there are 10 x 9 x 8 x 7 = 5040 ways to arrange 4 books out of a group of 10
Notice that this could be viewed as
If we let n = the total number of objects and r = the number chosen and arranged, then we could conclude that the number of ways to arrange n objects taken r at a time is
This can be quickly evaluated by MATH:PRB:2 (nPr)
Try it now on your calculator with n = 10 and r = 4

10. Example
How many three-letter “words” can be made from the letters A, B, C, and D?
You can use your calculator to answer this.
What are n and r in this problem?
Don’t worry that many of them are not real words; we don’t care in this context.
Write an organized list of all the possible “words”
Be systematic; be sure you write them all.

11. Three-letter “words”
ABC
ABD
ACD
BCD
ACB
ADB
ADC
BDC
BCA
BDA
DCA
CDB
BAC
BAD
DAC
CBD
CAB
DAB
CAD
DBC
CBA
DBA
CDA
DCB
Notice that being organized helps find all 24 permutations
Notice also that ABC is different from ACB because in permutation order matters
Suppose we don’t care about order. Then we are looking at combination, not permutation.
How many combinations of three letters can be made from an alphabet with four letters?

12. Three-letter “words” ABC ABD ACD BCD ACB ADB ADC BDC BCA BDA DCA CDB
BAC
BAD
DAC
CBD
CAB
DAB
CAD
DBC
CBA
DBA
CDA
DCB
When order does not matter, ABC is the same as ACB (etc.), so there are only 4 combinations in the 24 permutations.
They can be seen in the 4 columns
Notice that there are 3! (which is r!) permutations of each combination.
They can be seen in the 6 rows
So to get the number of combinations of n objects taken r at a time, divide permutations by r!
The formula is
The calculator command is MATH:PRB:3 (nCr) Try it now.

13. Examples
How many ways are there to form a 3 member subcommittee from a group of 12 people?
How many ways are there to choose a president, vice-president, and secretary from a group of 12 people?

14. More Examples
There are 5 cabins in the woods at a certain vacation spot. Each cabin has a path that leads to each of the other cabins. How many paths are there in all?
This is the combination of 5 things taken 2 at a time (order does not matter), so 5C2=10
There are 100 communications satellites orbiting earth. Each satellite needs a transmit and receive channel to talk to each of the other satellites. How many channels are needed?
This time AB is different from BA, so use permutation: 100P2=9900
How many pentagons can be drawn from the vertices of a regular 13-gon?
Combination: 13C5=1287

15. Objectives
Use organized lists and tree (branching) diagrams to list all possible outcomes of a trial.
Identify whether permutation or combination is appropriate to count the number of outcomes of a trial.
Use formulas or calculator commands to evaluate permutation and combination problems.

16. Homework
Write answers to each problem and be prepared to share them with the class; I will randomly choose 3 students for each problem.
Create and solve a problem involving the permutation of n things taken r at a time
Create and solve a problem involving the combination of n things taken r at a time