Five. (20 points) Fill in the right table with $1,2,3,4,5,6$ respectively, so that in each row, the number on the left is less than the number on the right, and in each column, the number on top is less than the number below. How many ways are there to fill the table? Provide an analysis process.
Five, as shown in the right figure, from the known information we can get: $a$ is the smallest, $f$ is the largest, so $a=1$, $f=6$. From $bd$, we need to discuss the following two cases:
(1) When $bd$, then $b=3, d=2, c=4$ or
5.
In this case, there are the following two ways to fill:
\begin{tabular}{|l|l|l|}
\hline 1 & 3 & 4 \\
\hline 2 & 5 & 6 \\
\hline
\end{tabular}
\begin{tabular}{|l|l|l|}
\hline 1 & 3 & 5 \\
\hline 2 & 4 & 6 \\
\hline
\end{tabular}

In summary, there are 5 ways to fill.
(Provided by Lai Guangfa, Hefei No. 46 Middle School, 230061)