It is very important to check that we have not overlooked any possible outcome. One visual method of checking this is making use of a tree diagram.

Ex. Flip a coin, then roll a dieS = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

The Counting Principle: The total number of possibilities for two or more independent events is the product of the number of possibilities for each event.  Example: You want to create a computer code using the letters A, B, C, D, E, and F. If letters may be re-used how many possible codes are there to choose from?

At Johnny’s Burger Place, a customer can get a customized meal by ordering either a turkey burger, chicken burger, hamburger, or garden burger with a side order of potato chips or french fries with a choice of either juice, milk, or soda. •Use a tree diagram to list all the different combinations of a burger, side order and a drink. •Describe ways and give examples of how Johnny could change his menu so that a customer would have 30 different choices.

Permutations changing the order of elements arranged in a particular order. (ORDER MATTERS!) Example: Using the word BAT, how many three letter combinations can be made? (order matters and no letter may be repeated)   

10P3 = 10 • 9 • 8 = 720

Factorial (!) the product of a given positive integer multiplied by all lesser positive integers. This is a case of permutations where all of the objects are used.

Example: You want to create a computer code using the letters A, B, C, D, E, and F. This time letters may only be used once. How many possible codes are there to choose from?

1) If you have a combination lock that contains only the numbers from 0 to 9, and the combination contains three numbers, how many possible combinations exist for this lock (assume numbers can repeat)?  2) There are 7 books on a shelf. How many different ways can you arrange them?  3) How many different ways can we arrange the letters in the word MATH?

10 • 10 • 10 = 103 = 1000

7 • 6 • 5 • 4 • 3 • 2 • 1= 7! = 5040

4 • 3 • 2 • 1= 4! = 24

n P k =

n!

(n k)!n : total number of objects in a groupk : total number of objects taken from n

Example: If six divers are entered into a competition, how many possibilities are there for the top three places? (remember order matters!)

If 40 names are placed in a hat, how many permutations could be made if 15 names are selected? (assume order matters because of the different prized awarded) 

22

1540 1026.5!25

!40

!1540

!40

P

Combinations the arrangement of elements into various groups without regard to their order in the group. Example: Using the word BAT, how many two-letter combinations can be made? (remember order doesn’t matter!)   

n C k =

n!

k!(n k)!

n : total number of objects in a groupk : total number of objects taken from n

Example: With 32 seeds at Wimbledon (a famous tennis tournament in Europe), how many two player combinations are there for the final match?

816,983,13123456

444546474849

)!649(!6

!49649

C

1) How many different ways can you eliminate all of the 16 balls from a pool table (assuming that hitting the 8 ball in early doesn’t end the game like real pool)? Order matters!   

2) How many ways can first and second place be awarded to 10 people?   

16! = 2.09 • 1013

10P2 = 10 • 9 = 90

3) Using the word numbers: (a) If order matters, how many different arrangements are there for all letters in numbers?  (b) If b was definitely the first letter, now how many possible arrangements are there?  4) You have 5 shirts, but you will select only 3 for your vacation.  In how many different combinations of shirts can you bring? 

7! = 5040

6! = 720

10123

345

)!35(!3

!535

C