Say we have a bag of 16 balls, consisting of 8 black balls, and 8 white ones. Now we want to draw the balls from the bag one by one without replacement, but we also make sure that the number of black balls drawn are always more than or equal to that of white balls. The question is how many possible combinations are there for drawing balls?
The tricky part of this problem is that the state, i.e.the number of balls of each color drawn, between each draw is obviously dependent, which makes the counting complex. Let's denote the number of balls remains in the bag as N(b,w). Now the starting state is N(8,8), and it could only be N(7,8) in next. But after that it could be N(7,7) or N(6,8). And this goes on without a clear rule for the change of state. Counting from N(7,7) is similar to counting from N(8,8), which makes this problem kind of recursive. But I don't really know how to deal with counting from N(6,8).