Probability and Counting Rules 4-4: Counting Rules

Counting Rules Many times a person must know the number of all possible outcomes for a sequence of events. To determine this number, three rules can be used. Fundamental Counting Rule Permutation Rule Combination Rule

The Fundamental Counting Rule Example: If a woman has three skirts and four sweaters, how many outfits are possible. Answer: skirts has 3 possibilities = k 1, sweaters has 4 possibilities = k 2. k 1  k 2 = 3  4 = 12.

Example of Fundamental Counting Rule

EX: What if repetitions are not allowed?

Example of Fundamental Counting Rule Suppose the state of Michigan has a new license plate style. The new license plates will have three letters followed by three numbers. Assuming that repetitions are allowed, how many license plates could be issued? How many license plates could be issued if repetitions are allowed?

Factorial Notation Factorial notation uses an exclamation point, ! Example: Calculate 5! Example: Calculate 9!

Permutations A permutation is an arrangement of n objects in a specific order. The calculation of permutations uses factorials. Example: You have four cars in your driveway, how many different ways can you line up the four cars in your driveway? This is a permutation since you are ordering the four cars.

Example of Permutations

In this example, she is not using up all 5 locations, she is only ordering 3 of them. “Out of 5, she is only choosing 3.” (We will learn a formula for this.)

Permutation Rules Think of this as ordering n objects, choose r.

Example of Permutation Rules

Permutation Rules In the previous examples, all items involving permutations were different, but when some of the items are identical, a second permutation rule can be used.

Example of Permutation Rules Example: Mrs. Cottrell has 9 old yearbooks on her shelf, 4 are from 2015, 2 are from 2014, 1 is from 2013 and 2 are from How many different ways can she order the yearbooks on her shelf?

Example of Permutation Rules

Combinations A selection of distinct objects without regard to order is called a combination. This is different from a permutation because in a combination, order DOES NOT MATTER. The difference between a permutation and a combination can be seen in a set of four letters {A, B, C, D} where two are chosen. Permutations {AB},{BA},{AC},{CA},{AD},{DA},{BC},{CB},{BD},{DB},{CD},{DC } Combinations {AB},{AC},{AD},{BC},{BD},{CD} order mattersorder does not matter {AB} and {BA} are the same combination. {AB} and {BA} are different permutations.

Combinations Combinations are used when the order or arrangement is not important, as in the selecting process. Example: Choose 4 students from our class to represent the class at a Statistics conference. It doesn’t matter who is chosen first, second, etc. We just want to choose 4 in any order…

Example of Combinations Example: Choose 4 students from our class to represent the class at a Statistics conference. It doesn’t matter who is chosen first, second, etc. We just want to choose 4 in any order…

Example of Combinations Notice that…

Example of Combinations

Summary of Counting Rules