Counting and Probability It’s the luck of the roll.

Factorial 3! = 3 x 2 x 1 = 6 5! = 5 x 4 x 3 x 2 x 1= 120 X! = x(x-1)(x-2)…1 3! Is read as “3 factorial” On the TI-83 the factorial key is located by pressing MATH arrow over to PROB down to 4:!

Arrangements When we put things in order it is important to know if repetition is allowed. How many 3 digit codes can be made using the numbers 0 – 9 if repetition is allowed? If repetition is not allowed?

Examples How many license plates can be made using 3 digits and 3 letters, repetition is allowed? How many license plates can be made using 3 digits and 3 letters if the numbers cannot repeat? How many 7 character passwords can be made using letters and numbers, repetition is allowed?

Fundamental Counting Principal Assignment You are the ruler of a small kingdom of 50,000 peasants, each owning a vehicle. Your ego urges you to design a new license plate showing your face as the background. Using numbers, letters, and/or symbols, what would be on your license plate so that there will be enough for every citizen and allow for 8% growth during your 4-year rule?

Menu Madness Using the menu from a local eatery, create combination offerings that could be “red plate specials.” From what items would you have them choose and what would be the cost? List the possible selections for your specials.

A permutation is an ordered arrangement. The number of permutations for n objects is n ! n ! = n ( n – 1) ( n – 2)… The number of permutations of n objects taken r at a time (r  n) is: Permutations

Example You are required to read 5 books from a list of 8. In how many different orders can you do so? There are 6720 permutations of 8 books reading 5.

A combination is a selection of r objects from a group of n objects. The number of combinations of n objects taken r at a time is Combinations

Example You are required to read 5 books from a list of 8. In how many different ways can you choose the books if order does not matter. There are 56 combinations of 8 objects taking 5.

2 3 Combinations of 4 objects choosing Each of the 6 groups represents a combination

2 3 Permutations of 4 objects choosing Each of the 12 groups represents a permutation

Example In a race with eight horses, how many ways can three of the horses finish in first, second and third place? Assume that there are no ties.

Example The board of directors for a company has twelve members. One member is the president, another is the vice president, another is the secretary and another is the treasurer. How many ways can these positions be assigned?

More Menu Madness Panda Express offers 3 entrees plus a side for a given price. What will you offer? Rather than have your eatery’s special require people to pick from different categories, you’re allowing them to pick any 3 things from any where on the menu. What might the cost be? How would you write this selection option?

Distinguishable Permutations The number of distinguishable permutations of n objects where n 1 are one type and n 2 are of another type and so on, is: n! n 1 !n 2 !n 3 !… n k !

Example To find the number of permutations in the word Mississippi total letters = 11! each set of repeats 2!4!4!

Example A contractor wants to plant six oak trees, nine maple trees, and five poplar trees along a subdivision street. If the trees are spaced evenly apart, in how many distinguishable ways can they be planted?

Example The manager of an accounting department wants to form a three- person advisory committee from the 16 employees in the department. In how many ways can the manager do this?

Making It Personal Write Casa Grande on your paper. How many distinguishable permutations are possible? Try Sacaton. Write out your entire given name. How many distinguishable permutations are possible using the letters in your name?