Basic Counting Problem I was reading a probability book and am having trouble conceptually with one of the examples. The following is a modification. 
Let's say that we have $3$ coins that we want to randomly assign into $3$ bins, with equal probability. We can label these coins $a_1$, $a_2$, $a_3$. What is the probability that all $3$ bins will be filled?
The solution is: All possible combinations of assigning these coins to bin locations is $3^3 = 27$. The possible ways that all 3 bins can be filled is $3!$. The final probability is $6/27 = 2/9$. Alternatively this could be derived as $(3/3)\cdot(2/3)\cdot(1/3) = 2/9$. 
Now what if the coins are not labeled and are considered interchangeable. There are now $10$ configurations in which these bins can be filled: $\binom{3+3-1}{3} = 10$. Only one of these configurations will have all bins filled. Thus the probability here is $1/10$. 
Shouldn't these probabilities be the same? Am I missing something with the second scenario?

It depends on whether you're thinking of choosing each arrangement of balls with equal probability, or whether you place the balls in each bin with equal probability. I would say that the most natural way to think about this is the latter, since this is most likely how it would happen in real life. 
When dealing with events that are independent like the placement of the unlabelled balls, I would think about it as placing one first, then another and finally a third. This is kind of a way to artificially label the balls, that makes intuitive sense. In the real world, you would probably place the balls one after another. At the very least, you would decide which one goes where one after another. If you did decide on the arrangement of them all at once, the chances are you would not be doing so independently and the outcome you describe with probability $\frac{1}{10}$ would occur.
This idea of making events happen one after another is extremely useful throughout probability (actually, it is useful in combinatorics, to decide how many ways there are of doing things much more easily by splitting up a complicated scenario into smaller ones - it applies to probability when we have events happening with equal probability, and then the probability of an event occuring is the number of ways it can happen divided by the number of total events which could happen.)
To bring it back to your example, assuming the balls are placed independently we can say that, if we want to fill up all three bins, the first one definitely goes into a bin, the second one goes into an empty one with probability $\frac{2}{3}$ and the third one goes into the final empty one with probability $\frac{1}{3}$ giving total probability of the event $\frac{2}{9}$
If, however, we assume that each arrangement is equally likely, then we get the answer $\frac{1}{10}$. This is because some arrangements are more likely to happen then others when the balls are placed independently, so the probabilities get skewed accordingly. For example, we have already established that the event of them all being in different bins can happen in $6$ different ways. The event of them all being in the same bin can only happen in $3$ different ways because there are $3$ different bins, so it should occur with half the probability of the balls all being spread evenly. This is what causes the discrepancy in the answers.
