[tex]\displaystyle\\
\frac{123!}{121!}=\frac{121! \times 122 \times 123}{121!}= 122 \times 123 = \boxed{\bf 15006}\\\\\\
\frac{(n+1)!}{n!} = \frac{n!\times(n+1)}{n!} =\boxed{\bf n+1}\\\\\\
\frac{(n+3)!}{n! \times (n+3)}=\frac{n! \times(n+1)(n+2)(n+3)}{n! \times (n+3)}=\boxed{\bf (n+1)(n+2)}\\\\\\
\frac{(n+1)! \times n!}{(n-1)!\times(n+2)!}= \frac{n!}{(n-1)!}\times \frac{(n+1)!}{(n+2)!}=\\\\
= \frac{(n-1)!\times n}{(n-1)!}\times \frac{(n+1)!}{(n+1)! \times (n+2)}=
\boxed{\bf \frac{n}{n+2}}
[/tex]
