Simple combinatorics (counting) problem

Question

How many ways are there to put 6 balls in 3 boxes if the balls are not distinguishable but the boxes are?

Answer 1

We have $6$ balls and imagine $2$ sticks that we will use as separators. For instance, the arrangement

$\bigcirc$ $\bigcirc$ $\bigcirc$ $\mid$ $\bigcirc$ $\mid$ $\bigcirc$ $\bigcirc$

is the one where you place 3 balls in the first bin, 1 ball in the second bin, and 2 balls in the third bin. How many ways can we arrange these items where the balls and sticks are indistinguishable? It is $\frac {8!}{6!2!}$. This gives you the answer.

(The sticks are indistinguishable because if you consider any arrangement and interchange the positions of the two sticks, you still have the same distribution of the balls)