Five people have applied for three different positions in a store. In how many ways can the positions be filled? 
Five people have applied for three different positions in a store. If each person is qualified for each position, in how many ways can the positions be filled?

Can someone tell me if I have to use permutations $\mathrm P(5, 3)$ or combinations $\mathrm C(5, 3)$ and why?
Thanks for any help!
 A: The normal way is to see that since the positions are distinct, order matters, and thus it is a permutation in which there are $5\cdot4\cdot3$ ways to successively fill the $3$ distinct posts.
Another way to look at it is :
Suppose you choose $3$ people, in $\binom53$ ways.
You still have to allot them their distinctive positions in $3!$ ways,
and note that $\binom53\cdot 3!$ is precisely the same as $^5P_3$
Either way, you will see clearly that you need to use permutations.
A: Well, consider the positions available. There are three. They literally told you that they are different. So consider positions one two and three.
You have five candidates. You fill the first position and there are 5 ways to do it. 
Then you consider the second position. You already hired one person, so there are 4 people left. Hence, there are four ways to fill position two.
Lastly, there are 3 ways to fill spot three.
Hence, there are $5\cdot 4\cdot 3$ ways to fill the positions.
Secretly, we used permutations. There are are 3 positions that are different and hence "the order matters". There are five candidates. Hence ${_5\mathsf P}_3$.
A: Label the positions are $P1, P2$ and $P3$. Remember they're different, so that could be, say, operations manager, human resources, and logistics.
Now, if you use combinations, the order in which you obtain a result does not matter. So if $A, B, C, D$ and $E$ applied (say, Alice, Bob, Carol, David and Edward), and you got $ABC$ or $CAB$ that would be the same.
However, here order does matter. When you get a permutation, assign the first person to the first job, the second person to second job, and the third person to third job.
So, for our previous example, $ABC$ would correspond to Alice getting operations manager, Bob getting human resources and Carol getting logistics. This is different than $CAB$, which amounts to Carol getting operations manager, Alice getting human resources and Bob getting logistics.
The gist of it is that, since order matters, permutations is the way to go.
