Brainteaser: What's my 10-digit number?

I'm thinking of a 10-digit number, where each digit represents the number of that ordinal number in the whole number. So, the first digit represents the number of 0's in the whole 10 digits. The second digit represents the number of 1's in the whole 10 digits. And so on. The first digit is not a '0.' What is my number?

Answer:
- Since the first digit is not zero, there is at least one zero among the digits.
- The sum of all digits must be 10 because it is a 10-digit number.
By some trials we find that the number 6210001000 fits the bill.

Follow-up question: is it the only correct answer?

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