Distributing identical objects to identical boxes

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 ?

Solution 1:

Distributing identical objects to identical boxes is the same as problems of integer partitions.

So if the objects and the boxes are identical, then we want to find the number of ways of writing the positive integer $n$ as a sum of positive integers. That is, if we consider the integers a sequence of positive integers $(a_1,a_2,\dots,a_k)$ such that the sum of those $a_i$ (for all $i$) sum to $n$, then the sequence $(a_1,a_2,\dots,a_k)$ form a partition on $n$. Note: $(1,3)$ is the same as $(3,1)$

You could:

1. Count by hand if the integer is small enough (as in this case). (See Aziumut's solution)
2. Use a Ferrer's diagram. As a hint there's a "faster" way to count those partitions if you make use of some of the relevant theorems.

Solution 2:

The result is the number of partitions of 6 into a sum of 4 integral summands $\geq 0$.

The possibilities are $$ 6 = 6 + 0 + 0 + 0 \\ = 5 + 1 + 0 + 0 \\ = 4 + 2 + 0 + 0 \\ = 4 + 1 + 1 + 0 \\ = 3 + 3 + 0 + 0 \\ = 3 + 2 + 1 + 0 \\ = 3 + 1 + 1 + 1 \\ = 2 + 2 + 2 + 0 \\ = 2 + 2 + 1 + 1 $$

so the answer is $9$.