Q1. Rephrase: how many ways can we place 5 caps in 5 boxes, if each box has 1 cap. Suppose the boxes represent positions; we can think of this question as how many ways can we order 5 caps? This is simply 5! = 120 ways.

Q2. "at least one cap has to be distributed" suggests that we could arrange all 5 caps as in Q1, but we could also use just 4 caps, or just 3, 2, or 1 cap.

If we find out how many ways we can place 1 cap, then 2, 3 and 4 caps, we can add up all those ways to 120 (ways to place 5 caps) to find out all possible outcomes.

Each of these ways can be broken into 3 steps.
(1) select the caps,
(2) select the boxes that will contain the selected caps,
(3) find # of ways to arrange the selected caps into the selected boxes

Multiplying (1) x (2) x (3) will give us all the ways to place the selected number of caps into boxes

Let's try this out

Ways to place 1 cap in 5 boxes.
(1) there are 5 ways to pick 1 cap from 5.
(2) there are 5 ways to pick 1 box from 5.
(3) there is only 1 way to arrange 1 cap in 1 box.
So there are 5 * 5 * 1 = 25 ways to use 1 cap. This should make conceptual sense since there are 5 ways we can put a cap in a box, and 5 caps we can pick from.

Ways to place 5 caps in 5 boxes.
(1) there is 1 way to pick 5 caps from 5.
(2) there is 1 way to pick 5 boxes from 5 to contain the caps
(3) there are 5! = 120 ways to arrange 5 caps in 5 boxes.
So there are 1 * 1 * 120 = 120 ways to use 5 caps. This is consistent with what we found from Q1 above.

Ways to place 2 caps in 5 boxes.
(1) there are 5!/(2!3!) = 10 ways to pick 2 caps.
(2) there are 10 ways to pick 2 boxes to contain the caps
(3) there are 2! = 2 ways to arrange the 2 caps in the 2 boxes (1st cap in 1st box, 2nd cap in 2nd box or reverse)
So there are 10 * 10 * 2 = 200 ways to use 2 caps.

Ways to place 3 caps in 5 boxes.
(1) there are 10 ways to pick 3 caps
(2) there are 10 ways to pick 3 boxes to contain the caps
(3) there are 3! = 6 ways to arrange the 3 caps in the 3 boxes.
So there are 10 * 10 * 6 = 600 ways to use 3 caps.

Ways to place 4 caps in 5 boxes.
(1) there are 5!/(4!1!) = 5 ways to pick 4 caps (makes sense; 5 ways to choose the cap that does NOT get picked)
(2) there are 5 ways to pick 4 boxes to contain the caps.
(3) there are 4! = 24 ways to arrange the 4 caps in the 4 boxes.
So there are 5 * 5 * 24 = 600 ways to use 4 caps.

So if we consider all possible selections (we can pick any number of caps from 1 to 6), there are 25 + 120 + 200 + 600 + 600 = 1545 possible outcomes