Fundamental of Counting principle

[joi]

Hello. I am looking for a confirmation on some problems I am working on. The first is: Applying the fundamental of counting principle, In how many different ways can a police department arrange eight suspects in a police lineup if each lineup contains all eight people?

Well, since there are eight people in the line up, and each lineup contains eight people, by the principle, the number of ways is equal to 8 * 8 = 64. Is this correct?

The second is: You need to arrange nine of your favorite books along a small shelf. Applying the fundamental of counting principle, How many different ways can you arrange the books, assuming that the order of the books makes a difference to you.

I believe that since there are nine books and only one shelf, the number of ways is equal to 1 * 9 = 9. Is this also correct?

Thank you for any insight.

 

[pka]

The way you have written the statement it is very hard to follow your meaning.
In any queue of eight people there are 8!= 40320 ways to arrange the people.
The first can be chosen in 8 ways, the second in 7 ways, the third in 6 ways, etc.
So you get by the basic counting rule, 8!=(8)(7)(6)…(2)(1)= 40320.

 

[JakeD]

Hi. You're not seeing the fundamental counting princlple correctly. Let's try a simpler example. Suppose there are 3 books on a shelf, numbered 1, 2, 3. How many ways are there to arrange the books?

The principle says if you have a task with 3 stages, the total number of choices is (choices in first stage task) times (choices in second stage task) times (choices in third stage task).

So for 3 book example, first stage task is choose the first book. There are 3 choices.

Second stage task is choose second book. After first book is chosen, there are only 2 left. So there are 2 choices for second task.

Third stage task is choose third book. But there is now only one book left. So only 1 choice for third task.

Total number of choices is \(\displaystyle \L 3 \times 2 \times 1 = 6.\)

Listing out the 6 ways to rearrange:

123
132
213
231
312
321

For the lineup problem, you have an 8-stage task like this one. You need to multiply 8 numbers together.

For the bookshelf problem, there are 9 stages, so multiply 9 numbers.

 

[joi]

Fundamental Counting Principle

Thank you for your help!!

Joi