# Thread: Ambiguity in probability questions

1. ## Ambiguity in probability questions

I feel that many probability questions are very ambiguous concerning the choice of words. For example:

In a talent show there are 12 contestants. The judges will choose 5 contestants and place them in the top 5 places. How many ways can this be done?

I seem to stumble upon these sorts of questions all the time where I find myself asking: ''Do they want me to find out how many ways 5 contestants can be placed in 5 places?'' ''Do they want me to find out how many ways 5 contestants can emerge from 12 and then find out how many ways these 5 can be placed in the top 5 places?''.

Maybe it's because I lack probability knowledge but is there any way to understand these explicitly?

2. ## Re: Ambiguity in probability questions

I think it's mostly just practicing these types of questions.

In your example, we know there are 12 total contestants. We need to pick 1st place, 2nd place, and so on.

How many choices of people do we have for first place? Answer: 12
For second place we have 11 choices
then 10 for 3rd, 9 for 4th and 8 for 5th.

So the overall answer is: $12 * 11 * 10 * 9 *8$ (or $12 \;P\; 5$)

We can also arrive at the answer a different way:

First we choose 5 out of the 12 people $\binom{12}{5}$ (or $12 \;C\; 5$)
then we rank them: 5 choices for 1st, 4 for 2nd, 3 for 3rd, 2 for 4th, 1 for 5th ( $5!$ choices total)

So the answer we get is $\binom{12}{5} * 5!$

If you check on your calculator, you will see that these two seemingly different answers are the same.