i encountered a problem on counting.

we have a 3 dimensional space. starting from (0,0,0), we need to go to (a,b,c) where a, b and c are positive integers by adding (1,0,0), (0,1,0) or (0,0,1). how many different paths are there?

using a tree diagram, i tried to find out the number of paths of a=1,b=1,c=1 and a=1,b=2,c=1. 6 and 12 respectively.

from those, i wrote an equation which do not look alright.

total number of moves n = a + b + c

number of possible paths = nCa * (n-a)Cb * (n-a-b)Cc

what is wrong for my equation? (or is it actually correct? )

thanks in advance.