apply the counting principle and permutations lesson 10.1 algebraii
TRANSCRIPT
Apply the Counting Principle and Permutations
Lesson 10.1AlgebraII
Agenda
Lesson 10.1Homework 10.1; On pages 686-689 do exercises 1, 4, 5, 9, 11, 16, 18, 23, 28, 34, 38, 43, 48, 63.
Apply the Counting Principle and Permutations
The fundamental counting principle:Two events.. If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur is .
DEFINITION SLIDE
Apply the Counting Principle and Permutations
• You are a member of the math team and the team has 3 different t-shirts that they wear in competition and 2 different coordinating pairs of sweatpants. The team wants to have a different uniform for every competition that they have. How many different competitions can the team attend without repeating a uniform?
• By the counting principle… 2x3=6• EXAMPLE SLIDE
Apply the Counting Principle and Permutations
The fundamental counting principle:Three or more events.. The fundamental counting principle can be extended to three or more events. For example, if three events can occur in m, n and p ways, then the number of ways that all three events can occur is .
DEFINITION SLIDE
Apply the Counting Principle and Permutations
We can also make a tree diagram for this problem:
DEFINITION SLIDE
Apply the Counting Principle and Permutations
• You are a member of the math team and the team has 3 different t-shirts that they wear in competition and 2 different coordinating pairs of sweatpants. The team wants to have a different uniform for every competition that they have. How many different competitions can the team attend without repeating a uniform? Now let’s add 5 hats to the mix…
• By the counting principle… 2x3x5=30• EXAMPLE SLIDE
Apply the Counting Principle and Permutations
If you look at California plates you will see that they start with a single digit, then are followed by 3 letters then followed by three numbers. The letters I,O and Q are may not be used in the 1st or 3rd position on the plate… How many unique standard plates can be issued before California needs to go to a new system…There are 10 digits available and there are 26 letters in the alphabet.EXAMPLE SLIDE
Apply the Counting Principle and Permutations
10x23x26x23x10x10x10… that’s a lot!
137,540,000
There are about 25 million cars registered in the state of California.
SOLUTION SLIDE
Apply the Counting Principle and Permutations
Now let’s talk about permutations…Remember Johanna from the Pizza parlor… she likes ice cream cones. The store that she visits has 24 flavors of ice cream. It’s important to her which flavor scoop is on top. After all, she says, eating chocolate and then vanilla is a different taste experience from eating vanilla and then chocolate. She always orders two different flavors.How many different two-scoop ice cream cones can Johanna create?EXAMPLE SLIDE
Apply the Counting Principle and Permutations
Let’s use the counting principle to find out…
This type of counting is called a permutation… Sometimes it’s necessary to carry this calculation out quite far so we need to have an easier way.
Apply the Counting Principle and Permutations
Introducing (pronounced n factorial). This is equal to
The number of permutations of n distinct objects is given by Interesting fact… DEFINITION SLIDE
Apply the Counting Principle and Permutations
Do problems 9, 13 and 15 from Practice B
Apply the Counting Principle and Permutations
We can also use to help us calculate the ice-cream cone problem…
EXAMPLE SLIDE
Apply the Counting Principle and Permutations
Definition: Permutations of n Objects taken r at a time
The number of permutations or r objects taken from a group of n distinct objects is denoted by
DEFINITION SLIDE
Apply the Counting Principle and Permutations
For our example of the ice cream cones…
EXAMPLE SLIDE
Apply the Counting Principle and Permutations
Do problems 17, 19 and 24 from Practice B.
Apply the Counting Principle and Permutations
We can also look at the different permutations when we have repetitions… The letters a, a and b. If the two a’s are how many different ways can we arrange these letters to form unique words?
Apply the Counting Principle and Permutations
The letters a, a and b. If the two a’s are how many different ways can we arrange these letters to form unique words?aab aba baa… so how do figure this out.. If we had three different letters it would be 3x2x1 because of the number of permutations.
What process can we find to do this systematically?
Apply the Counting Principle and Permutations
aab aba baa… so how do figure this out.. If we had three different letters it would be 3x2x1 because of the number of permutations.
What process can we find to do this systematically?
EXAMPLE SLIDE
Apply the Counting Principle and Permutations
Formally..
Permutations with repetition: The number of distinguishable permutations of objects when one object is repeated times another is repeated times, and so on, is:
DEFINITION SLIDE
Apply the Counting Principle and Permutations
Find the number of distinguishable permutations of the letters in the word:1. MALL2. KAYAK3. MISSISSIPPI
EXAMPLE SLIDE
Apply the Counting Principle and Permutations
SUMMARY SLIDE:1. Counting principle2. Tree diagram
3. Permutations4. Permutations with repetition