N balls in r urns Could someone please explain the difference between these two cases:
1. No. Of ways to arrange n distinguishable balls in r urns.
2. No. Of ways to arrange n balls in r urns when the balls are identical.
It would be great if someone could explain them taking specific values of n and r.
 A: Hint: We should also consider if urns are distinguishable or not. If this is not explicitly stated, we typically assume they are indistinguishable (but should state this assumption in an answer).  Here is an example with $n=3$ balls and $r=4$ urns.

We think of $N=\{1,2,3\}$ as a set of balls and of $R=\{a,b,c,d\}$ as a set of urns. A function $f:N\rightarrow R$ is considered as  placing each ball into some urn.
We consider four functions $j,k,l,m: N\rightarrow R$ by
  \begin{array}{lclcllcl}
j(1)&=&j(2)&=&a,&\qquad j(3)&=&b\\
k(1)&=&k(3)&=&a,&\qquad k(2)&=&b\\
l(1)&=&l(2)&=&b,&\qquad l(3)&=&d\\
m(2)&=&m(3)&=&b,&\qquad m(1)&=&c\\
\end{array}

Four functions with distinguishable balls and urns:
                                 
with balls indistinguishable:
                                 
with urns indistinguishable:
                                 
with balls and urns indistinguishable:
                                 
Note: Some more information is given in this answer.
A: In the first case, let's make it simple and consider 2 balls and 2 urns. Have the balls be labeled X and Y, respectfully. Since in the first case the balls are not identical, this means that the probability of choosing X is not equal to choosing Y. The these events are not equally probable, no matter in what urns they are pulled from. In the second case, we have balls X and Y, which are identical. This means that the probability of choosing X then Y is the same as choosing Y then X afterwards. The key thing to remember is the word identical. 
A: Both the balls and the urns are distinguishable
$n=1,r=1$ gives $1!$ ways
$n=2,r=2$ gives $2!$ ways
$n=6,r=10$ you have a choice of selecting 6 urns from 10 and then put the balls in them that gives you $10\choose6$.$6!$   
In case of balls being identical you can disting which ball went in to which urn so if $n=6,r=10$
the answer is just $10\choose 6$
