I've never done a question like this in perms and coms (haven't even started the topic yet). There are two parts of the question and I don't know where to start!
We start from $A$. We are only allowed to move seven blocks exactly. We can only move in the directions northward and eastward.
a) How many possible routes are there from $A$ to $B$?
b) How many possible routes are there from $A$ to $B$, but you must go past $C$ before $B$?
$(a)$ Let move sequence be a binary string where $1$ denotes upward moves and $0$ denotes rightward moves. Then this binary string has length seven and it includes four $1$'s and three $0$'s. For example, $1111000$ or $0001111$. So part $(a)$ asks how many such binary strings are there and answer to this question is $$\binom{7}{4} = 35$$ because we are choosing the places for $1$'s and there are $7$ places in total. When we choose them, rest will be $0$ so actually it is $\binom{7}{4}\binom{3}{3} = \binom{7}{4}$.
$(b)$ Since it must pass from $C$, let us consider this problem in two parts. In the first part, let us find the ways from $A$ to $C$: With a similar logic to part $(a)$, there are $\binom{4}{3} = 4$ ways to go $C$ from $A$. Now, since we are in $C$, let's find the number of ways of going $B$ from $C$. This is $\binom{3}{2} = 3$. Therefore, we have $4 \cdot 3 = 12$ distinct ways.
EDIT: I read the comments and since the numbers are not so big in this case, you can consider part $(b)$ as the following:
Here, consider each curve as a different path in original question. Then you will eventually find the number of ways of going $B$ from $A$ in the above figure.