Proving a probability transition matrix I have the following question:

With the necessary show that proof:

My solution, states 'shown in class' but we were never taught how to derive this. I would really appreciate some help on this.
Thanks in advance.

As mentioned in the problem let X_n be the number of molecules in container A after n units of time have passed. 
Now clearly the state space is S={0,1,...,N} since at some point in time, container A can have all the molecules or none of the molecules. 
Now we can consider, by homogeneity of the problem what happens if X_0=0, that is there are 0 molecules in A. If this is the case then once the aperture opens, only one molecule can go from B to A, so P(X_1=1|X_0=0)=1 in terms of transition probabilities. Now consider what happens if X_0=1 (i.e we have 1 molecule in A). In this case once the aperture opens, one molecule (out of N) can go from A to B with probability 1/N or one molecule (from N-1 possibilities in B) can go from B to A with probability (N-1)/N. So P(X_1=0|X_0=1)=1/N (molecule goes from A to B) and P(X_1=2|X_0=1)=(N-1)/N (molecule goes from B to A). 
Now continue with this process to get the remaining transition probabilities noting that if there are N molecules in A, then only one molecule can go from A to B once this aperture opens. 
