Answer:
To solve this problem, we need to carefully analyze the sequence of numbers given and identify the instances where the number 7 is preceded by 3 and followed by 8.

The given sequence of numbers is: 382372833782837873787382

Step 1: Identify the instances where 7 is preceded by 3 and followed by 8.
We need to check each number in the sequence and identify if it meets the given criteria.

- 382: The number 7 is not preceded by 3, so we can ignore this instance.
- 823: The number 7 is not preceded by 3, so we can ignore this instance.
- 238: The number 7 is not preceded by 3, so we can ignore this instance.
- 837: The number 7 is preceded by 3 and followed by 8, so we count this instance.
- 373: The number 7 is preceded by 3 and followed by 3, not 8, so we can ignore this instance.
- 738: The number 7 is preceded by 3 and followed by 8, so we count this instance.
- 388: The number 7 is not preceded by 3, so we can ignore this instance.
- 883: The number 7 is not preceded by 3, so we can ignore this instance.
- 837: The number 7 is preceded by 3 and followed by 8, so we count this instance.
- 787: The number 7 is not preceded by 3, so we can ignore this instance.
- 873: The number 7 is preceded by 3 and followed by 3, not 8, so we can ignore this instance.
- 738: The number 7 is preceded by 3 and followed by 8, so we count this instance.
- 382: The number 7 is not preceded by 3, so we can ignore this instance.

Step 2: Count the instances.
From the analysis above, we have identified three instances where the number 7 is preceded by 3 and followed by 8. Therefore, the correct answer is option 'C' - 3.

Summary:
In the given sequence of numbers, the number 7 is preceded by 3 and followed by 8 three times.