Post on 19-Jan-2018
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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