How many ways to go from $(0,0)$ to $(20,10)$ if precisely $2$ right moves need to be made in a row? Just to clarify:  This is the number of ways to go from point $(0,0)$ to point $(20,10)$ if the only directions allowed are right and up. The catch: Each of the ways must include precisely (only) $1$ instance of a "double right".
I know that all of the ways to go from $(0,0)$ to $(20,10)$ can be found by $C(30,20)$ or $C(30,10)$, but I'm having trouble wrapping my head around how to only include ways that have precisely $1$ double right.

Imagine that your 'long right' was instead a single rightward step.  Then, rather than going from $(0,0)$ to $(20,10)$ instead you would move from $(0,0)$ to $(20-k+1,10)$, where $k$ is the length of the 'long right' — but you would do it without any two consecutive rightward moves.  What's more, since there's only one 'long right', we can replace any of the $j=20-k+1$ rightward moves by it and get a unique configuration (no two such configs can collide); this means that for each way of getting from $(0,0)$ to $(j, 10)$ without consecutive rightward moves, there are $j$ ways with one 'long right' move.  So once we have the quantity $C_j$ of such ways, we can calculate the sum $\sum_kjC_j$ over all $k$ to get the final answer.  In fact, since every value of $k$ corresponds to a unique value of $j$ and we never directly use the value of $k$, we may as well make our sum be over $j$ instead of $k$, and we'll consider it to be summed that way from here on out.
Now, let's look at that quantity $C_j$. Since now between any two rightward moves we know that there's at least one upward move, we can eliminate the first upward move between each of the $j-1$ pairs of consecutive moves to get a configuration where we instead have $j$ rightward moves and $10-(j-1)$ upward moves with no restrictions on them; contrariwise, corresponding to any one of these configurations of $j$ rightward moves and $10-(j-1)$ upward moves we have a unique configuration of $j$ rightward moves and $10$ upward moves by just re-inserting the $(j-1)$ upward moves we took out, one between each pair of rightward moves.  (You may have seen this under the name 'Stars and Bars' at some point).  Can you take it from here to see what $C_j$ must be, and then finish by calculating the original sum?
