Probability-How many ways can 5 finalists be ranked from 1st place to 5th place? I just need to ask this question because I really do not know what to do
For 5 finalists in a beauty contest, in how many ways can they be ranked from 1st place to 5th place?
I am really having a problem in probability and i hope that you guys can recommend some sites that contain lessons of this topic cause I am planning on studying the topic this coming weekend.. thank you in advance guys.  
 A: This is an application of one of the basic counting principles: if you perform $m$ tasks in succession, and there are $n_1$ different ways to perform the first task, $n_2$ different ways to perform the second, and so on, then there ar $n_1n_2\ldots n_m$ different ways to perform the entire string of tasks. This is called the rule of product, multiplication principle, or Chinese menu principle.
In your problem there are $5$ ways to pick the winner. Once the winner has been picked, there are just $4$ ways to pick the second-place finisher. When both of the first two finishers have been picked, any one of the remaining $3$ women can be the third-place finisher. At that point only two contestants remain, so there are $2$ ways to pick the fourth-place finisher, and of course that leaves only one choice for the last-place finisher. Multiplying these together, we get a total of $$5\cdot4\cdot3\cdot2\cdot1=5!=120$$ different possible finish orders.
The same reasoning shows that in general there are $n!=n(n-1)(n-2)\cdots(2)(1)$ different arrangements of $n$ distinct items.
I can’t recommend any particular on-line tutorials, because I’m not sufficiently familiar with them. A Google search on basic probability and counting will give you a lot of choices to examine. This PDF looks like a decent brief survey of material closely related to this problem.
A: Perhaps it may be useful to consider fewer finalists and see if there is a pattern that emerges:
If there is just 1 finalist, then there is only one way to rank that person. Thus 1 way in this case.
If there are 2 finalists, then there are a couple of ways to pick who is first and the other person is second.  Thus, 2 ways in this case.
If there are 3 finalists, then there are 3 ways to pick 1st place, 2 ways to pick second and whoever is left is going to be third.  So, that is 3 times 2 times 1 which is 6 ways in this case.
If there are 4 finalists, then there are 4 ways to pick 1st place, 3 ways to pick second place, 2 ways to pick 3rd place and whoever is left is 4th place.  This is 24 ways in total.
Now, do you see enough of a pattern to deduce how to handle the case of 5 finalists?
