To determine the number of distinct arrangements for wearing five different rings on the five fingers of the right hand, we need to consider the following:

1. Understanding the problem:
- We have 5 different rings.
- We have 5 fingers.
- Each ring can be placed on any of the 5 fingers.
- Multiple rings can be worn on the same finger.

2. Applying the multiplication principle:
- For each ring, there are 5 choices (one for each finger).
- Since the placement of each ring is independent of the others, we multiply the number of choices for each ring.

3. Calculating the total number of arrangements:
- For the first ring, there are 5 choices.
- For the second ring, there are 5 choices.
- For the third ring, there are 5 choices.
- For the fourth ring, there are 5 choices.
- For the fifth ring, there are 5 choices.

Therefore, the total number of distinct arrangements is:
$$5 \times 5 \times 5 \times 5 \times 5 = 5^5$$

4. Verification:
- The order in which the rings are placed on the fingers does not matter in this context because each ring is independently placed on any of the 5 fingers.
- The formula $\binom{5}{1} \cdot 5 \cdot \binom{4}{1} \cdot 4 \cdot \binom{3}{1} \cdot 3 \cdot \binom{2}{1} \cdot 2 \cdot \binom{1}{1} \cdot 1$ is not applicable here because it assumes a different context where the order of selection and placement is considered differently.

The final answer is $\boxed{5^5}$.