As Lucian commented, this question can be approached with partitions:
The partitions (order immaterial) of 7 number 15.
Also, at most, 7 can be partitioned into 7 integers (1+1+1+…). Also, fortunately, $n<10^8$ has 8 "slots" (the question would be a bit different if, say, the cap was $10^5$)
Now, some additional combinatorics comes in.
For any given partition of 7, say 4 + 2 + 1, there will be a corresponding amount of $0$s (as you need to fill up all 8 "slots".
Hence for the case of 4, 2, 1 and an additional 5 $0$s, we need to figure out the number of combinations, as this represents all the different integers up to $10^8$ with the said digits in them.
So we would have: $8!/(1!*1!*1!*5!)$
(there is the link to multinomials!)
Repeat this process for the 14 other partitions of 7, and there you have your answer