NEW UNIT = COMBINATORICS

AHH FACTORIALS!



Well I guess its my turn for scribe again. Anyway, today in class we learned about Factorials. If you don't know what a factorial is, I hope this post will help you learn. :)

First we started with a problem to help us ease into the new unit.

What we did here is find the number of possibilities 3 coins will land on a table. As you can see in this first part there is 2 possibilities which are heads and tails. When you toss 2 coins the possibilities double, and then at three times it doubles again to get 8 ways for these coins to land on the table.

*NOTE* This is not a factorial, just a way to ease us into this subject as I said before xD

Here is another slide in class.
How did we get this you may ask? In class Mr.K made 5 kids in the class get up to show the example.(I was one of them :S) It went like this:

Imagine the letters as 5 people. When the first person goes to sit down, he has 5 possible spots he can sit in. Which leaves 4 seats left for the next person to choose from. Every time a person takes a seat the possible spots to sit in decreases. Still don't understand? I'll show what I mean below:


Remember "A" can go into any of the 5 seats at the beginning, so if "A" went to another seat it would create a whole new possibility or even if any other letter was to change its spot. So each letter has a certain amount of possibilities.
We multiply all these numbers to find the total amount of possibilities that are possible. We do not want to keep finding the possibilities of such problems this way because they might become long and time consuming. Plus there is a faster way :D.

Learning the principle of counting will guide you to understanding more :D.


Self explanatory if you read all the text in the box :P but I'll explain it anyways. This principle states that if you have the # of ways one thing can do and the # of ways another can do, you can simply multiply them together to find the total amount of ways they both can do. (almost exactly what it says xD)

NOW FOR FACTORIALS!!! yay!



You might wonder, What do the exclamations mean? You might think this number is special. Well it is but not like a SUPER FANTASTIC ULTRA MEGALICIOUS NUMBER or something xD. The exclamation does not mean" Yes I mean this number!!!!!" If there is an exclamation beside a number in math, it just means that you take that number and count down till you get to one. Then you multiply all those numbers to get a result. Look below for a formula :D


By looking at this formula carefully you can see how you find the value of any factorial. 0! is still a mystery to why it equals 1, since Mr.k has not gave us that speech yet :D.

Here's a couple slides that might help you understand different factorial and counting uses :)


As you can see on the above slides, you can divide two different factorials by splitting it into what it is.
What we did on the left is break up n! into n(n-1)! which are the exact same thing. Also on the right we broke up (n+1)! into (n+1)n! which are also the same thing.

So (n+1)! = (n+1)n!


Now for a few questions we also did in class which might be challenging.
The 1st, 2nd, and 6th digits are given a number which will give only one possibility.
For the third digit it says its even. So this means that all even numbers from 0-9 could be in this spot. So that gives us 5 possibilities for the 3rd digit(0,2,4,6,8).
On the 4th digit it must be greater then 5. So that gives us 4 remaining numbers of 6,7,8,9 that could be here. Which is 4 possibilities.
Now in the 5th and 7th digit it says this number is odd. Which gives 5 possibilities(1,3,5,7,9).

We multiply all these possibilities to get the number of phone numbers that can be made under these certain conditions. Which turns out to be 500.

Now we did one more set of questions to end the class.

a) What we did was you have 7 letters but only have 4 slots to put those letters. So you start from 7 possibilities and go down by one each time. But since there is only 4 slots you have to stop at 3. Then you multiply these 4 numbers to get an answer of 840.

b) What you do first is look at the consonants and vowels. There are 4 vowels and 3 consonants. Which gives 4 possibilities for the 1st slot and 3 for the last one. When finding these middle numbers you must remember how many letters you have left after these slots are filled. Which gives you 5 and then 4. After multiplying all these 4 numbers together your should get 240 different words you can spell.

c) Last one is very interesting. You have two slots for vowels and two for consonants. Since you have 4 vowels to choose to place in the first slot. Next is a consonant so you put one out of the three here. Now you only have 3 vowels to choose from for the third slot and 2 consonants to choose from for the last slot. So that gives 4*3*3*2 amount of four letter words. BUT you must account for the possibility of this order being the other way around. What if you do Consonants first. So with the 72 you got from before, add it with the new value if it was the other way and that will equal the total amount of 4 letter words that can be created.


WHEW!!!!!!!!!!

That was a lot to write. I'm sorry if I complicated things or did not describe as well as I could. Just message me about any errors or complains. Sorry for the late post.

FINALLY!!!!

ITS

TIME

FOR

THE

NEXT

SCRIBE


ALEX

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