Putting $n$ Things in $m$ Boxes A bicycle collector has $100$ bikes. How many ways can the bikes be stored in four warehouses if the bikes and the warehouses are considered distinct? What if the bikes are indistinguishable and the warehouses distinct?
 A: Line up the bikes in order of age, or beauty, or serial number. The first bike can go into any of the $4$ warehouses, so we have $4$ choices. For every decision about the first bike, there are $4$ decisions we can make about the second bike. So the fates of the first two bikes can be decided in $4^2$ ways.
Continue. For every decision about the fate of the first two bikes, there are $4$ choices for where the third bike goes, and so on. So there are $4^{100}$ ways to do the whole job. 
You can think of an assignment of bikes to warehouses as a function from the set of bikes to the set of warehouses. There are $4^{100}$ such functions.
For indistinguishable bikes, we have a classical Stars and Bars problem. Perhaps you could look at the Wikipedia article. It is pretty good. This sort of question has also been answered many times on MSE.
A: For the first problem, we count arrangements like:A: 1,2,...,99  B: none C: 100 D: none
Each of the bikes has four places it can go, and these choices are all independent.  By the multiplication principle we count as $$4\times 4\times\cdots\times 4=4^{100}$$
For the second problem, we count arrangements like:A: 99 bikes  B: none C: one bike D: none
You are looking for a weak composition of $100$ into 4 parts.  The formula for this is $${n+k-1\choose k-1}={103\choose 3}$$
