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10-substrings

Published on Saturday, 10th October 2015, 10:00 pm; Solved by 174;
Difficulty rating: 85%

Problem 529

A 10-substring of a number is a substring of its digits that sum to 10. For example, the 10-substrings of the number 3523014 are:

  • 3523014
  • 3523014
  • 3523014
  • 3523014

A number is called 10-substring-friendly if every one of its digits belongs to a 10-substring. For example, 3523014 is 10-substring-friendly, but 28546 is not.

Let T(n) be the number of 10-substring-friendly numbers from 1 to 10n (inclusive).
For example T(2) = 9 and T(5) = 3492.

Find T(1018) mod 1 000 000 007.