Probability and Counting Rules4-4: 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 = k1, sweaters

has 4 possibilities = k2. k1k2 = 34 = 12.

Example of Fundamental Counting Rule

Example of Fundamental Counting Rule

Example of Fundamental Counting Rule

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

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 RulesThink of this as ordering n objects, choose r.

Example of Permutation Rules

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 2012. 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 matters order 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

Example of Combinations

Summary of Counting Rules