There are two ways of doing this. One is Ross Millikan's: you will make ten "up" moves, and 20 "right" moves; the only question is which order you make them in. Imagine placing the "right" moves on a row; now you need to decide where to do the "up" moves: you do so by inserting them "in between" (or before, or after) the "right" moves. So you need to choose ten places to put "up" moves: there are 21 locations for them (nineteeen in between the "right" moves, one before all of them, one after), and you are allowed to choose the same location more than once.
This is a combinations-with-repetitions: the formula is $\binom{n+r-1}{r}$, where you have $n$ possibilities, and must make $r$ choices with repetitions allowed. In this case, $n=21$, $r=10$, so you get $\binom{30}{10}$.
There is another way of doing it, which is more graphical. I'll do it with a 4 by 3 array so you see how it works. You have this array:
$$\begin{array}{cccc}
\cdot & \cdot & \cdot & \cdot\\
\cdot & \cdot & \cdot & \cdot\\
\cdot & \cdot & \cdot & \cdot
\end{array}$$
Now, you start at the bottom left, so there is only one way to get there; we put a $1$ next to it.
$$\begin{array}{llll}
\cdot & \cdot & \cdot & \cdot\\
\cdot & \cdot & \cdot & \cdot\\
\cdot\;1& \cdot & \cdot & \cdot
\end{array}$$
Then, you can go either up or right; there is only one way to get to those points (via the first move); we put a $1$ next to them:
$$\begin{array}{llll}
\cdot & \cdot & \cdot & \cdot\\
\cdot\;1 & \cdot & \cdot & \cdot\\
\cdot\;1& \cdot\;1 & \cdot & \cdot
\end{array}$$
Now: to get to $(1,1)$, you can either get to it from $(1,0)$ or from $(0,1)$; since there is only one way to get to each of those, there are two ways to get to $(1,1)$. On the other hand, only one way to get to $(2,0)$ or to $(0,2)$:
$$\begin{array}{llll}
\cdot\;1 & \cdot & \cdot & \cdot\\
\cdot\;1 & \cdot\;2 & \cdot & \cdot\\
\cdot\;1& \cdot\;1 & \cdot\;1 & \cdot
\end{array}$$
Next: to get to $(1,2)$, you can arrive either from $(0,2)$ (one way of being there), or from $(1,1)$ (two ways of getting there); so in total, three ways. Likewise, you have three ways to get to $(2,1)$, because you can either go up from $(2,0)$, and there is only one way to do all of that, or you can go right from $(1,1)$ (and there are two ways of doing that, corresponding to the two ways there are to get to $(1,1)$; so we have:
$$\begin{array}{llll}
\cdot \;1 &\cdot\;3 & \cdot & \cdot\\
\cdot\;1 & \cdot\;2 & \cdot\;3 & \cdot\\
\cdot\;1& \cdot\;1 & \cdot\;1 & \cdot
\end{array}$$
Continuing this way, we get:
$$\begin{array}{llll}
\cdot\;1 & \cdot\;3 & \cdot\;6 & \cdot\;10\\
\cdot\;1 & \cdot\;2 & \cdot\;3 & \cdot\;4\\
\cdot\;1 & \cdot\;1 & \cdot\;1 & \cdot\;1
\end{array}$$
So there are $10$ ways to get to the top right corner in the 4 by 3 case.
You may even recognize that these numbers are just Pascal's triangle lying on its side! Well, there is a combinatorial formula for the entries of Pascal's triangle: the $r$th entry in the $m$th row corresponds to the coefficient of $a^{m-r}b^{r-1}$ in the binomial expansion of $(a+b)^{m-1}$, so it equals $\binom{m-1}{r-1}$. To figure out the entry that corresponds to the top right corner, note that you go "down" one row for each position on the $x$-axis, and another one for each step up. So here we have gone to the 4th row on the horizontal steps, and then to the 6th row on the by going up twice. Each step up is a move "right" on the row. So with a 4 by 3, we are in row $4+(3-1)=6$, and we are in position $3$ of that row. According to the formula above, that corresponds to $\binom{4+2-1}{3-1}=\binom{5}{2}$. This corresponds to the need to make $3$ moves right and two moves up, so we need choose where to place the two up moves among the three right moves; there are four places to put them in (before the three, after the three, or in the two spaces in between), so the formula I gave above gives this answer as well.
I am assuming that you can only move down and right.
So label the inner corner, say $A_1 = (W-B,H-A)$, and count off $A_2 = (W-B-1,H-A-1)$, $A_3 = (W-B-2,H-A-2)$, until $A_k = (W-B-k, H-A-k)$, where $k = \min\{W-B,H-A\}$. Then, all your desired paths from top left corner $(0,H)$ to bottom right corner $(W,0)$ must pass through exactly one $A_i$ for some $i$. So, we want to count the number of paths from top left corner to $A_i$ and from $A_i$ to bottom right corner, multiply the two numbers (the product signifies the number of paths through $A_i$), and sum over $i$ to get the desired answer.
Best Answer
Observe that if you first move right, you must end with a move to the right, and if you first move down, you must end with a move down. In the former case, you are left with a grid of $6 \times 6,$ which has $12 \choose 6$ valid moves. In the latter case, you are left with a grid of $8 \times 4,$ which has $12 \choose 8$ valid moves. The number of ways to achieve an even number of direction changes thus equals:
$${12 \choose 6} + {12 \choose 8} = 924 + 495 = 1419$$