Tutorial On Counting Principals for Probability

Author: Will W.

This is a basic tutorial on Permutations and Combinations
If you don't know what they are. They're actually some of
The coolest parts of math that you might ever learn.

The first part will go over The Multiplication Principal
And permutations following with Combinations. I figure if
you wanna know about computers, and how the elite actually
break some of the toughest combinations this should be your
first start. I will go over several examples. I will not teach
you how to crack a code but if your smart enough you might
actually find out how to apply this to other sequences of
combinations.



So let’s get started with the Multiplication Principal

Suppose n choices must be made, with
m1 ways to make choice 1

And for each of those ways
m2 ways to make choice 2

and so on, with
Mn ways to make choice n.

Different ways to make the entire sequence of choices


Example 1:

A certain combination lock can be set to open to any one 3-letter
sequence. How many such sequences are possible?

Solution:
Since there are 26 letters in the alphabet, there are 26 choices
for each of the 3 letters. By the multiplication principal, there
are 26 X 26 X 26 = 17,576 . So there are 17,576 different sequences


That was a pretty basic example lets do one with Morse code.

Example 2:

Morse code uses a sequence of dots and dashes to represent letters
and words. How many sequences are possible with at most 3 symbols?

Solution:

-------------------------------------------------------------------
| Number of Symbols Number of Sequences |
--------------------------------------------------------------------
| 1 2 |
| 2 2 X 2 |
| 3 2 X 2 X 2 |
|__________________________________________________________________|

So, 2 + 4 + 8 = 14
That means there are 14 different sequences

You with me so far? It's not that difficult to grasp. We will do another
example of this. Just to get you more confused hehe.

Example 3:
my Sister has 5 different plants the he wishes to arrange side by side
on her window. How many different arrangements are possible to line up
the plants on her window?

Solution:
Since there are only five plant that means only five choices will be
made. After that then there will be 4 choices left. Then after that
there will be 3 choices left and so on and so on. So do find the answer
you simply just do this:
5 X 4 X 3 X 2 X 1
which is 120. So there are 120 different arrangements my sister could
arrange her plants.

Alright that was the last example. Pretty easy stuff right? Well I hope
your still with me. Now let’s move on. Before we move on we need to learn
F A C T O R I A L Notation yipee!!!.

so lets get some syntax here.
! means a factorial, that's how you identify a factorial
0! = 1 means it's only 1 duh

Now for any natural number n
n! = n(n-1)(n-2) ...(3)(2)(1)

Now basically what were doing is saying that 5 X 4 X 3 X 2 X 1 is 5!
So 7!(7fractorial) is the same as 7 X 6 X 5 X 4 X 3 X 2 X 1. If you have
a calculator with the symbol n! this is what that means. The calculator
is gonna come in very handy.

And this is how to work the equation. Lets give n! = 5!

so n! = n( n - 1) (n - 2)
5 = 5( 5 - 1) X ( 5 - 2)
5 = 5 X (4) X (3)

You would keep plug in ( n - 3) until you got to 1
which is the same thing basically as 5 X 4 X 3 X 2 X 1
Alright so you all should be with me so far. Lets move on to permutations

Permutations:
Sounds cool don't it? I think it does. Now Just follow along.
This is the basic equation for permutations.

If P(n,r) (where r < n) is the number or permutations on (n) elements
-
taken r at a time then
P(n,r) = n!
---
(n - r)!

A permutation is an arrangement of items where the ORDER of the
arrangement matters.

Lets do an example
example 1:
A baseball team has 20 players. How many 9-player batting orders are possible?

Solution:
Now lets use the equation
n is going to be 20 so n = 20
r is going to be the 9 player batting order so r = 9
So lets use the equation of permutation.

P(n,r) = n!
---
(n - r)!

Simplify
P(20,9) = 20!
----
(20 - 9) !

Simplify
20!
----
(11)! which would equal 60,949,324,800.

So the solution to the example would be there are 60,949,324,800 different orders
that are possible. Get it? Well maybe next time you will be nicer to the coach
when he messes up the batting order. Just shout out coach it's not your fault
there are 60,949,324,800 different combinations. Who knows you might actually
get a new job.


example 2:
In how many ways can the letters in the word Mississippi be arranged?

Solution:
The word contains 1 m, 4 i's, and 2 p's. To use the formula, let
n = 11, n1 = 1, n2 = 4, n3 = 4, n4 = 2 to get

11!
---------
1!4!4!2!

Which is equal to 34,650 combinations. So pretty cool right. Makes you
sound smart! yippeee!

Now lets move onto combinations. Combinations are the best. But they take a little
more work since combinations not care about an order they are arranged.

Basic Equation:

If (n) (it's one big parenthesis)
(r)

If (n) denotes the number of combinations of n elements taken r at a time
(r)
where r < n, then
-
( n ) = n!
( r ) ----------
(n - r)!r!

Another frequently used notation is

C(n,r) (which is the same and the easiest)

SO lets do another example! whoa

example 1:

Lets do something with Poker. How many different combinations can we have when playing with 5 cards in one hand?

Let's Break this down.

n = number of cards in a deck so 52
n = 52

r = number of cards were playing with
r = 5

Use the equation C(n,r) = n!
-----
(n -r)!r!

Simplify:
C(52, 5) = 52!
------
(52 - 5)!5!
Simplify:
C(52,5) = 52!
------
( 47)!5!

You should get 2,980,000. So that shows there are 2,980,000 different combinations that are gonna be used. If you go further in depth you can actually figure out the combination of face cards, the number of full houses, just by applying the combination equation. This is great for playing with dominoes.





Well anytime you need to find the exact combination and sequence of arrangements just apply this to your problem. The combination is used alot more in daily life. For example back in the early 90's 4 millionaires corned the Power Ball lotto on a 10 million jackpot. They only spend around 2 million dollars on tickets making a great investment. I hope you enjoyed reading this tutorial. It's a great way to sound smart and impress girls, guy's whatever.