Step 1: Understand the setup.
We have 500 dissimilar boxes and 50 distinct persons. Each person must receive exactly $ \frac{500}{50} = 10 $ boxes.
Step 2: Think of it as sequential group assignments.
First, give 10 boxes to person 1. Then from the remaining 490, give 10 to person 2. Continue until all boxes are distributed.
Step 3: Count ways for each person.
Person 1 gets $ {}^{500}C_{10} $ choices. Person 2 gets $ {}^{490}C_{10} $ choices. Person 3 gets $ {}^{480}C_{10} $ choices, and so on.
Step 4: Write out the product.
Total = $ {}^{500}C_{10} \times {}^{490}C_{10} \times \cdots \times {}^{10}C_{10} $. This equals: \[ \frac{500!}{10! \cdot 490!} \times \frac{490!}{10! \cdot 480!} \times \cdots \times \frac{10!}{10! \cdot 0!} = \frac{500!}{(10!)^{50}} \]
Step 5: Explain why we do NOT divide by 50!.
The 50 persons are distinguishable (they are different people). If the boxes were divided into unlabeled groups, we would divide by $ 50! $. But since each person is distinct, no extra division is needed.
Step 6: State the final answer.
\[ \boxed{\dfrac{500!}{(10!)^{50}}} \]