This is equivalent to asking how many ways there are to use 3 numbers from the set of {0, 1, 2, 3, 4, 5, 6} (repeats allowed).
Ways:
6,0,0
5,1,0
4,1,1
4,2,0
3,3,0
3,2,1
2,2,2
Identical balls: It doesn't matter what order the balls go in.
Identical boxes: It doesn't matter which box the balls go in. In the list of ways, this is equivalent to saying that 3,2,1 is the same as 3,1,2 or 2,1,3, etc..
7 ways.