Distinguishability in Probability theory Reading Sheldon and Ross Introduction to Probability Theory. I'm on section 1.6. 
They say that there are $r^{n}$ possible outcomes when n distinguishable balls are to be distributed among r distinguishable urns. This makes sense to me. 
Then they say that the answer is different when the $n$ balls are indistinguishable. I take it indistinguishable means that the balls(for all intents or purposes) are representing the same exact object. Then they say that it is a vector ($x_1$,$x_2$,...,$x_r$).
I don't understand what's happening in this case. How is the answer different if the balls are exactly the same?
 A: Lets assume that we line up r distinguishable urns in a line. Urn A, Urn B, ... Urn R.
The question starts off by saying that there are $r^n$ different ways to distribute n distinguishable balls into r distinguishable urns. Because each urn can hold more than one ball, each ball has r different places it could end up, thus $r^n$
Now, lets suppose that the balls are indistinguishable (i.e. for all intents and purposes the balls are exactly the same). This would mean that when we distribute 4 balls into 3 urns, that if we put 1 ball in urn A 1 ball in Urn B and 2 balls in urn C, we count this only one time.(It doesn't matter on which distribution the ball ended up in urn A, a ball in urn A and a ball in urn B is counted only once. This is opposed to distinguishable balls and urns) 
This turns the problem more into a vector problem. Now the goal is to the find the distinct vectors $(x_1,x_2,...,x_r)$ = $n$.
Thus the number of solutions to this problem is almost clear. It basically says how many different ways can you create r groups from these n indistinguishable objects. The book uses a clever answer which uses combinations to find the answer.
0 | 0 |... | 0 | 0 
Let all the 0 be the balls. The question than becomes, how many different ways can you separate these balls. There are n-1C r-1 (because if you are creating 3 urns you need to separators. This thus gives us that there are that many positive distinct solutions to the the equation $x_1 + x_2 + ... + x_r = n$.
