This is $\color{red}{\text{ “indistinguishable objects into indistinguishable boxes(IOIB)” }}$ problem which is a standard combinatorial problem. There is no simple closed formula for the number of ways to distribute $n$ indistinguishable objects into $j$ indistinguishable boxes.
So, We will enumerate all the ways to distribute.
Best way is to go in a sequence, covering all possibilities, So, that we do not overcount, we do not undercount.
Case 1 : $\color{blue}{\text{If only one bin is used : }}$
(6,0,0) i.e. Only 1 way (Since All balls have to be put in this single bin that is used ; Since all bins are identical, so, it doesn’t matter which bin we use)
Case 2 : $\color{blue}{\text{If two bins are used : }}$
The distribution can be done as any of the following :
(5,1) (which means 5 identical balls in one bin, 1 ball in another bin, and the third bin is unused)
(4,2) (which means 4 identical balls in one bin, 2 balls in another bin, and the third bin is unused)
(3,3) (which means 3 identical balls in one bin, 3 balls in another bin, and the third bin is unused)
i.e. 3 ways to distribute 6 identical balls into 3 identical bins if exactly two of the bins are used.
Case 3 : $\color{blue}{\text{If three bins are used : }}$
The distribution can be done as any of the following :
(4,1,1) (which means 4 identical balls in one bin, 1 ball in another bin, and 1 ball in the third bin)
(3,2,1) (which means 3 identical balls in one bin, 2 ball in another bin, and 1 ball in the third bin)
(2,2,2) (which means 2 identical balls in one bin, 2 ball in another bin, and 2 ball in the third bin)
i.e. 3 ways to distribute 6 identical balls into 3 identical bins if all three of the bins are used.
So, total we have 7 ways to distribute 6 identical balls in 3 identical bins.
NOTE that we cannot distribute as (2,2,1,1) because only three bins are available, not four.
$\color{red}{\text{Learn ALL about IOIB Here, with Variations:}}$ https://www.youtube.com/watch?v=Vyhp6fvaoso&list=PLIPZ2_p3RNHgm_UqwqckMxM68HS4BkjYY&index=34