MHB What are the different ways to place two objects in five slots?

[tmt1]

I'm getting these concepts confused.

If I have an object called $x$, and I have five places or slots to put the object, how many ways could 2 $x$s be places in the 5 spaces?

Example:

x x _ _ _
x _ x _ _
x _ _ x _
x _ _ _ x
_ x x _ _
_ x _ x _
_ x _ _ x
_ _ x x _
_ _ x _ x
_ _ _ x x

So in this example there are 10 ways to place the $x$s (am I missing any?).

So would this be a combination with repetition, permutation or something else, and what formula can I use to calculate this?

 

[MarkFL]

This would be a combination, since order doesn't matter. Here you are simply looking to find how many ways there are to choose 2 from 5:

$$N={5 \choose 2}=\frac{5!}{2!(5-2)!}=10$$

You have 5 choices for the first x and 4 choices for the second, and by the fundamental counting principle, this is $5\cdot4=20$ ways to place the two x's, but since the two x's are identical, then the order doesn't matter, so we have to divide by the number of ways to order the 2 x's which is $2!=2$, and so we find we have 10 different placements. If the two things you are placing into the 5 slots are different, say you are going to place an x and a y, then order would matter and there would be 20 permutations. :D