2100010006 is the only 10 digit decimal number string such that "1st digit indicates the number of 1's in the number, the 2nd indicates the number of 2's in the number and so on... till the 10th indicates the number of zeros in the number."

With the help of an Excel spreadsheet, it took me a couple of minutes to find 2100010006, and another 30 minutes to eliminate all other possibilities.

Firstly, let us simplify the problem:

All digits have to add up to 10 (they represent the number of digits in a 10 digit long string)

Now let us concentrate on the most important digit, the one that represents the 0s and then try to come up with all possible combinations for this problem.

I've rearranged the order to make this a bit easier to visualise

9 8 7 6 5 4 3 2 1 0 where
0 0 0 1 0 0 0 1 2 6 is the solution

- For obvious reasons 0 column cannot be zero.
- There can not be nine 0s because we would need to put a one in the 9 column (reducing the number of 0s to 8)
- There can not be ANY 9s at all (you could try nine 1s, but that won't work)
- We will place 0s in the highest possible column, starting at 9 and working our way down
- When there are eight 0s, there can only be two digits 8 and 2 (that add up to 10)
- When there are seven 0s, there can only be three digits 7, 2 and 1
- When there are six 0s, there can only be four digits 6, 2, 1 and 1
- and so on... as represented by the below table

9 8 7 6 5 4 3 2 1 0
1 0 0 0 0 0 0 0 0 9 IMPOSSIBLE
0 0 0 0 0 0 0 0 x 8 must have 2 digits: 8+2 = 10
0 0 0 0 0 0 0 x x 7 must have 3 digits: 7+2+1 = 10
0 0 0 0 0 0 x x x 6 must have 4 digits: 6+2+1+1 = 10
0 0 0 0 0 x x x x 5 must have 5 digits: 5+2+1+1+1 = 10
0 0 0 0 x x x x x 4 must have 6 digits: 4+2+1+1+1+1 = 10
0 0 0 x x x x x x 3 must have 7 digits: 3+2+1+1+1+1+1 = 10
0 0 x x x x x x x 2 must have 8 digits: 2+2+1+1+1+1+1+1 = 10
0 x x x x x x x x 1 must have 9 digits: 2+2+1+1+1+1+1+1+1 = 10
x x x x x x x x x 0 IMPOSSIBLE

In the below table we will:
- Plug in some numbers into the top three rows with 7,8 or 9 0s; we start running out of 0s pretty quickly
- Plug in more numbers into the bottom 5 rows; results in the sum of all digits being above 10

9 8 7 6 5 4 3 2 1 0
1 0 0 0 0 0 0 0 0 9 IMPOSSIBLE (had to go down to 8 zeroes)
0 1 0 0 0 0 0 0 1 8 IMPOSSIBLE (had to go down to 7 zeroes)
0 0 1 0 0 0 0 1 2 7 IMPOSSIBLE (had to go down to 6 zeroes)
0 0 0 0 0 0 x x x 6 ANSWER has six 0s
0 0 0 0 1 0 1 1 3 5 IMPOSSIBLE (5+6=11)
0 0 0 0 1 1 x x x 4 IMPOSSIBLE (5+6=11)
0 0 0 1 1 x x x x 3 IMPOSSIBLE (5+6=11)
0 0 1 1 x x x x x 2 IMPOSSIBLE (4+7=11)
0 1 1 x x x x x x 1 IMPOSSIBLE (3+8=11)
x x x x x x x x x 0 IMPOSSIBLE

Now we just use brute force:

9 8 7 6 5 4 3 2 1 0
0 0 0 1 0 0 0 x x 6 there is one 6
0 0 0 1 0 0 0 x 1 6 there is one 1
0 0 0 1 0 0 0 x 2 6 now there are two 1s
0 0 0 1 0 0 0 1 2 6 now there is one 2, and BAM! we've stumbled upon the the only answer out of 10,000,000,000

This is a visual/logical proof that I came up with to see if there were any other numbers that matched the criteria, for a mathematical proof, please consult your local mathematician.