$\displaystyle\Pi_{i=1}^{n}{\frac{i}{i+4}}=\frac{1}{5}\cdot\frac{2}{6}\cdot\ldots\cdot\frac{n}{n+4}=\frac{1\cdot2\cdot3\cdot\ldots\cdot n}{5\cdot6\cdot7\cdot\ldots\cdot(n+4)}=\frac{n!}{\frac{(n+4)!}{\color{red}{4!}}}=\frac{n!\cdot4!}{(n+4)!}$
Why did it suddenly divide by $\color{red}{\textrm{factorial of } 4}$ during the transformations? If you think about it, the factorial of 4 is "missing" in the denominator due to the characteristics of the sequence. However, mathematically this action still makes sense. Please explain why the division took place, I thought about it myself but nothing comes to mind.