I will be talking about what I have learned this week in our precal class. First of all I would like to thank Mr. Piatek for introducing me to this "blogging site" that I have never heard of but I'm just glad that I have. It'll be a very interesting semester with him :).
Permutations Part 1
FUNDAMENTAL COUNTING PRINCIPLE
I will introduce the fundamental counting principle with an example.
This counting principle is all about choices we might make given many possibilities.
Suppose most of your clothes are dirty and you are left with 2 pants and 3 shirts.
How many choices do you have or how many different ways can you dress?
Let's call the pants: pants #1 and pants #2
Let's call the shirts: shirt #1, shirt #2, and shirt #3
Then, a tree diagram as the one below can be used to show all the choices you can make
As you can see on the diagram, you can wear pants #1 with shirt # 1. That's one of your choices.
Count all the branches to see how many choices you have.
Since you have six branches, you have 6 choices.
However, notice that a quick multiplication of 2 × 3 will yield the same answer.
In general, if you have n choices for a first task and m choices for a second task, you have n × m choices for both tasks
In the example above, you have 2 choices for pants and 3 choices for shirts. Thus, you have 2 × 3 choices.
Another example:
You go a restaurant to get some breakfast. The menu says pancakes, waffles, or home fries. And for drink, coffee, juice, hot chocolate, and tea. How many different choices of food and drink do you have?
There 3 choices for food and 4 choices for drink.
Thus, you have a total of 3 × 4 = 12 choices.
Tips and advice:
Multiplication Principle: Key Words are "AND" "BOTH"
- when two events are dependent we multiply their results to find the total number of ways an event can occur.
Another example would be:
If there are 52 runners entered in a race, in how many ways can first AND (indicates that we have to multiply) second place be awarded?
therefore: 52 x 51 = 2652 ways
-------- ---------
1st place 2nd place
So where does 51 come from? I am glad you asked :)
Well we have 52 runners in total and we can have any of these 52 individuals who can be awarded First, and since we can only have one first place winner, the other 51 individual can either place Second. And that is where 51 came from, and if you're not with me, Mr. Piatek can clarify the rest for you :)
Additive Principle: Key Words are "OR", "EITHER", "OPPOSITE"
- when two events are independent we add their results to find the total number of ways an event can occur.
An example would be:
Suppose that the executive of the Manitoba Association of Mathematics Teachers consists of three woman and two men. In how many ways can a president and a secretary be chosen if:
a) The president is to be female and the secretary male?
So here we have 3 Female and 2 Male:
3F = president
2M= secretary
therefore: 3 x 2
--------- ---------- = 6 different ways
President Secretary
b) The president is to be male and the secretary female?
So now we have 2 Male and 3 Female:
2M= president
3F= secretary
therefore: 2 x 3
-------- --------- = 6 different ways
President Secretary
c) The president and secretary are to be of the OPPOSITE (indicates that we have to add) sex?
So now it's asking for how many ways can the president and secretary are to be of the opposite sex, simple:
We have 3 females and 2 males, and we can name the females, Marjory, Lucy, and Gertrude and the males we can name them, John and Ash, just so it's easier for us to understand and not get mixed up.
Since there are 6 different ways for them to become a president and secretary we write our own, let's say columns to show the 6 ways.
therefore: 1st column 2nd column
President(F)/ Secretary(M) President (M)/ Secretary(F)
M J J M
M A J L
L J J G
L A A M
G J A L
G A A G
----------------------------- ----------------------------------
= 6 ways + = 6 ways
and 6+6 = 12
So now we add both of them and we get 12 different ways in total! :)
So far this is what I have in mind that I have learned over the week, and I hope you enjoyed browsing over my blog, and learning about Fundamental Counting Principles! Thank you!
Till next time ladies and gentlemen! Woo, Precal's fun and blogging! :)
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