Let's Think "SEQUENCE." Random Number Generators, Statistics, and Probability
* How many sequences can we make out of the number 3? Ans: One. The sequence is 3.
* How many seuqences can we make out of the numbers 1 and 1? Ans. One. The sequence is 11.
* How many sequences can we make out the numbers 2 and 3 if we don't care about the order? Ans: One. The sequence is 23 or 32. This is one sequence because we don't care about the order.
* How many sequences can we make out of the numbers 2 and 3 if we do care about the order? Ans: Two. The sequences are 23 and 32.
* How many sequences can we make out of the numbers 1,2, and 3 if we care about the order? Ans: Six.
It is easy to count them manually if we want, but we can also count them without counting by using the FUNDAMENTAL PRINCIPLE OF COUNTING, which says for this particular case that we have 3 tasks to do. We can do the first task in 3 ways, the 2nd task in 2 ways, and the third task in 1 way. Therefore, our answer is 3*2*1 = 6.
First Task:
The first task is to choose a number for one of the positions (It doesn't matter which position, but people usually choose the first). We can do this in 3 different ways.
Second Task:
The second task is to choose one of the other numbers for one of the positions still available (It doesn't matter which one, but people usually choose the second). We can do this in 2 different ways because there are 2 numbers left.
Third Task:
At this point there is only one number left and one position left. We can choose this number in one way.
Now we multiply them together. 3*2*1 = 6.
We multiply them together because there are 3 ways to choose the first number. Then for each one of these 3 ways, there are 2 ways to choose the next number. Then for each of these 2 ways, there is 1 way to choose the last number.
Listing our possibilities manually to check our answer, we get 123 132 231 213 312 321.
The Fundamental Principle of Counting is so important because we get into problems that are too difficult for us to list all the possible sequences.
For example, How many sequences can we make from the 10 different letters a,b,c,d,e,f,g,h,i,j ?
The answer is denoted by 10! (pronounced TEN FACTORIAL) = 10*9*8*7*6*5*4*3*2*1 = 3,628,800.
We can make more than 3 million different sequences.
But what if we don't care about the order of some of the characters of our sequence?
Consider the numbers 1,2,3, & 4.
We just learned that if we care about order, there are 24 different sequences we can form.
We denote this by 4! = 4*3*2*1 = 24
But what if we don't care about the order of these numbers? How many sequences can we form?
The answer is 1. This is because the order is not important, e.g., 1234 is in effect the same sequence as 4321.
Now, the BIG POINT I want to make here is "How do we do this mathematically?"
We DIVIDE OUT the order. 4*3*2*1 / 4*3*2*1 = 1. This is the same as 4! / 4! = 1.
Let's consider another example
Take the numbers 1,2,3,4, & 5. We know that we can form 5! different sequences if we care about the order of each element in the sequence.
But how many different sequences of length 2 can we form from these 5 different numbers?
Answer: 5*4*3*2*1 / 3*2*1 = 5*4 = 20
We have divided the order out of 3 of the positions.
If we think in terms of the FUNDAMENTAL PRINCIPLE OF COUNTING, we see that
there are 5 ways to do the FIRST TASK and 4 ways to do the SECOND TASK.
Therefore, we have 5*4 = 20 different possible sequences of length 2.
To Be Continued