The number of lattice paths from $(0, 0)$ to $(7, 7)$ What is the number of lattice paths of length $14$ from the point $(0, 0)$ to the point $(7, 7)$ such that they pass through the point $(4, 4)$ and don't pass the line $y = x$? 
Note: each step is right on the grid, or up on the grid.
I am thinking about using the Inclusion–exclusion principle, but I just don't know how to implement it here!

We may introduce Catalan's number from Bertrand's ballot problem.

Bertrand's ballot problem. There is a ballot between candidates $A$ and $B$ for the mayor election. At the end of the vote count $A$
  is the winner. Then the probability that $A$ has been ahead of $B$
  during the whole vote count is $\frac{A-B}{A+B}$.

Proof: It is enough to understand what are the chances of a tie at some point of the vote count. If the first vote is for $B$ that happens for sure. If the first vote is for $A$ but there is a tie at some point, by switching votes for $A$ and for $B$ till the tie we are in the previous situation. It follows that the wanted probability is $1$ minus twice the probability that the first vote is for $B$, i.e. $1-\frac{2B}{A+B}=\frac{A-B}{A+B}$.
If we assume that $A$ is the winner with $n+1$ votes, $B$ is the loser with $n$ votes and the first vote is for $A$, we get that in a $n\times n$ grid the number of paths from $(0,0)$ to $(n+1,n+1)$, made by steps towards north or east, that stay on or above the diagonal is
$$ \frac{1}{2n+1}\binom{2n+1}{n}=\frac{1}{n+1}\binom{2n}{n}=C_n.$$
With this preamble we immediately get that the answer is given by $2 C_3 C_4 = \color{red}{140}$.
These are our faboulous five for going from $(4,4)$ to $(7,7)$:

And these are our faboulous fourteen for going from $(0,0)$ to $(4,4)$:

