**Explanation:**

To find the Roman numeral for the greatest three-digit number, we need to understand the rules of Roman numerals.

**Roman Numerals:**

Roman numerals are a numeral system that originated in ancient Rome. They use combinations of letters from the Latin alphabet to represent numbers.

The basic symbols used in Roman numerals are:

- I represents 1
- V represents 5
- X represents 10
- L represents 50
- C represents 100
- D represents 500
- M represents 1000

Roman numerals are formed by combining these symbols together. However, there are some rules to follow when forming Roman numerals.

**Rules for Forming Roman Numerals:**

1. Symbols can be repeated up to three times in a row.
2. If a smaller symbol appears before a larger symbol, its value is subtracted from the larger symbol.
3. If a smaller symbol appears after a larger symbol, its value is added to the larger symbol.

**Finding the Greatest Three-Digit Number:**

The greatest three-digit number is 999. To represent this number in Roman numerals, we need to break it down into its digit values.

999 can be written as 900 + 90 + 9.

**Representation of 900:**

900 can be written as CM. Here, C represents 100 and M represents 1000. When a smaller symbol (C) appears before a larger symbol (M), its value is subtracted from the larger symbol.

**Representation of 90:**

90 can be written as XC. Here, X represents 10 and C represents 100. When a smaller symbol (X) appears before a larger symbol (C), its value is subtracted from the larger symbol.

**Representation of 9:**

9 can be written as IX. Here, I represents 1 and X represents 10. When a smaller symbol (I) appears before a larger symbol (X), its value is subtracted from the larger symbol.

**Combining the Representations:**

Now, we can combine the representations of 900, 90, and 9 to form the Roman numeral for 999.

CM + XC + IX = CMXCIX.

Therefore, option C (CMXCIX) is the correct Roman numeral representation for the greatest three-digit number.

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