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 $51C_5 = 2349060$.