\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
\mbox{(1)} \qquad C_{n}
 & = & \frac{(2n)!}{(1 + n) (n!)^{2}} \\
& = & \frac{(2n)!}{(1 + n)! n!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
C_{p}
 & = & \frac{(2p)!}{(1 + p)! p!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(2)} \qquad F_{n}
 & = & \frac{(3n)!}{(1 + 2n) (2n)! n!} \\
& = & \frac{(3n)!}{(1 + 2n)! n!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F_{q}
 & = & \frac{(3q)!}{(1 + 2q)! q!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(3)} \qquad H (p, q)
 & = & \frac{(2p + 3q)!}{(1 + p + 2q)! p! q!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
H (p, q)
 & = & \frac{(2p + 3q)!}{(1 + p + 2q)! p! q!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(4)} \qquad H (p, 0)
 & = & \frac{(2p + 3 \cdot 0)!}{(1 + p + 2 \cdot 0)! p! 0!} \\
& = & \frac{(2p)!}{(1 + p)! p!} \\
& = & C_{p} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(5)} \qquad H (0, q)
 & = & \frac{(2 \cdot 0 + 3q)!}{(1 + 0 + 2q)! 0! q!} \\
& = & \frac{(3q)!}{(1 + 2q)! q!} \\
& = & F_{q} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\mbox{(6)} \qquad H (h_{3}, h_{4}, h_{5}, h_{6}, \ldots, h_{n})
 & = & \frac{(2 h_{3} + 3 h_{4} + 4 h_{5} + 5 h_{6} + \ldots + (n - 1) h_{n})!}
{(1 + h_{3} + 2 h_{4} + 3 h_{5} + 4 h_{6} + \ldots + (n - 2) h_{n})! h_{3}! h_{4}! h_{5}! h_{6}! \cdots h_{n}!} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
H (1, 0, 1)
 & = & \frac{(2 \cdot 1 + 3 \cdot 0 + 4 \cdot 1)!}{(1 + 1 + 2 \cdot 0 + 3 \cdot 1)! 1! 0! 1!} \\
& = & \frac{(2 + 0 + 4)!}{(1 + 1 + 0 + 3)! 1! 0! 1!} \\
& = & \frac{6!}{5! 1! 0! 1!} \\
& = & \frac{720}{120 \cdot 1 \cdot 1 \cdot 1} \\
& = & 6 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
H (1, 0, 0, 1)
 & = & \frac{(2 \cdot 1 + 3 \cdot 0 + 4 \cdot 0 + 5 \cdot 1)!}{(1 + 1 + 2 \cdot 0 + 3 \cdot 0 + 4 \cdot 1)! 1! 0! 0! 1!} \\
& = & \frac{(2 + 0 + 0 + 5)!}{(1 + 1 + 0 + 0 + 4)! 1! 0! 0! 1!} \\
& = & \frac{7!}{6! 1! 0! 0! 1!} \\
& = & \frac{5040}{720 \cdot 1 \cdot 1 \cdot 1} \\
& = & 7 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
H (1, 2, 1)
 & = & \frac{(2 \cdot 1 + 3 \cdot 2 + 4 \cdot 1)!}{(1 + 1 + 2 \cdot 2 + 3 \cdot 1)! 1! 2! 1!} \\
& = & \frac{(2 + 6 + 4)!}{(1 + 1 + 4 + 3)! 1! 2! 1!} \\
& = & \frac{12!}{9! 1! 2! 1!} \\
& = & \frac{479001600}{362880 \cdot 1 \cdot 2 \cdot 1} \\
& = & 660 \\
\end{eqnarray*}
\end{document}
