TITLE: Distributing identical objects to identical boxes
QUESTION [6 upvotes]: We have 6 identical things to be distributed in 4 identical boxes such that empty boxes are allowed the find the number of ways to distribute the things ?

REPLY [2 votes]: Since all objects and boxes are identical, this means we don't care which object goes to which box, the only thing we care about is how many objects in each boxes, and this is equivalent to say how many ways can you partition a number $r$ into $k$ parts?
if we let $P_k(r)$ be the number of way to partition the number $r$ into $k$ parts, in your case $k=4$ and $r=6$, then we have to find how many ways we can partition 6 into 4 parts. But also keep in mind that there is allowing empty boxes, this means that we may also have to consider the partition of 6 in to 3 parts, 2 parts or 1 part, add up all cases which gives $$\sum_{k=1}^4P_k(6)$$
And yet, there is no "real" formula to calculate $P_k(r)$, but for small numbers you may count it by hand.