I suppose you have tried to summarize something much more detailed, which makes this hard to follow. As I understand it, you are reinterpreting the problem (about four-digit numbers) as a new problem where 4 children are dividing 8 indistinguishable balls among them; the first must have at least one ball, but the others may have none. (This is a partitioning problem.) You want to know how many ways this can be done, which is equivalent to the original problem. I don't see how the notation "8D 3C" helps. Then you are remodeling the problem yet again, as something unstated about distances between trees and pens. I can't follow that at all.
But I think what you are probably doing is what we call in English "Stars and Bars", because we remodel the problem in terms of figures like |***||**|****|, which represents either 4 "bins" (pens?) containing 3, 0, 2, 4 balls respectively, making the required total of 9; or alternatively just an arrangement of stars and bars, taking the figure literally. This gives us a total of 11 positions, if we ignore the two outer bars which are fixed, of which we choose 3 to be bars, so that the total is "11C3", which is the calculation you show. But then you would have to subtract the arrangements with nothing in the first bin.
Does this sound like what you were taught? If so, you can search for "stars and bars" to find information. Possibly "coding" means that, or perhaps it means the more general idea of remodeling a problem, changing its representation to one that can be solved.