Japan Mathematical Olympiad Initial Round

$$\boxed{2349060}$$ Consider a particle moving on the $xy$-plane according to the following rule: if the particle is at the point $(n,m)$ where $n$ and $m$ are integers, it moves in one step either to the point $(n+1,m)$ or to the point $(n,m+1)$. Consider the set $\Omega$ of all paths that a particle can take to start at the origin $(0,0)$ and take steps, following the rule as above to reach the point $(46,5)$. It is easy to see that there are exactly $51C_5$ such paths. Among these paths, there are exactly $17C_8\cdot 17C_8\cdot 17C_8$ (where $a+b+c=5$) paths which cross the lines $x+y=17$ and $x+y=34$ at the points $(17-a,a)$ and $(34-a-b,a+b)$, respectively. Since all the paths belonging to the set $\Omega$ cross these two lines at these points for some choice of $a$ and $b$ with $0\le a,b,$

and $a+b\le 5$, we see that the value of the sum we seek is exactly the same as the number of paths belonging to the set $\Omega$, which is $51C_5=2349060$.