Given eight distinguishable rings, let N be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. The number of distinct digits in N are

There are $\left(\begin{array}{c}8\\ 5\end{array}\right)$ ways to choose the rings, and there are 5! Distinct arrangements to order the rings [we order them so that the first ring is the bottom-most on the first finger that actually has a ring, and so forth]. The number of ways to distribute the rings among the fingers is equivalent the number of ways we can drop five balls into 4 urns, or similarly dropping five balls into four compartments split by three dividers. The number of ways to arrange those dividers and balls is just $\left(\begin{array}{c}8\\ 3\end{array}\right)$.
Multiplying gives the answer: $\left(\begin{array}{c}8\\ 5\end{array}\right)\left(\begin{array}{c}8\\ 3\end{array}\right)5!=376320$