Sum of 5 Puzzle

The Puzzle:

Sum of 5

A 5-digit number has the sum of all five digits equal to 5.

The first digit is equal to the number of zeros, the second digit is equal to the number of ones, and so on.

What is the number?

(Example that does NOT work: 30200 says "3 zeros, 2 twos", but there is only 1 two, so try again!)

Do you have the answer? Check against our solution!