I can see 2 ways to count the # of solutions. In both of these, I count from 0 to 999,999 and manually count the number of ways from 1,000,000 to 1,000,015.
First, list the number of ways to write 15 = 9+6 =8+7, = 5+5+5, etc. Then find the number of ways to put these digits into the 6 slots that form the numbers from 0-999,999. for instance, there are 6 choose 2 ways to put a 9 and a 6 into a list of 6 slots. The other slots are 0. This way seems a bit tedious.
The 2nd way is to consider the number 15 as 15 coins, and the division between digits as lines that you insert between coins. There should be 5 lines, making 20 items total. Then the number of ways to insert 5 lines into 20 slots is 20 choose 5. The resulting number is the number of coins between lines. For instance, line, coin, coin, line, line, coin is the number 0201. The bad point about this method is that you might have 15 coins between 2 lines, indicating a digit with value 15. There is of course no such digit base 10.