If it is correct, then we have the following problem: we must ultimately find the number of ways of distributing 6 white and 8 black marbles to the 3 boys in such a way that every boy gets at least 1 marble (presumably, either black or white), but there is also the condition that we do not have to distribute all of the marbles we got.

Therefore, we must give away at least 3 marbles (either black or white) -so that every boy gets one- and at most all 14 marbles.

I know we should subtract some quantities from the total number of distributions (13860), but am not really sure how precisely to go about that...