A Counting Problem 2

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Points: 15
Time limit: 2.0s
Memory limit: 64M

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Problem type

You are given a digit array \(a\) of length \(N\). You must find the number of integers between \(0\) and \(K\) (inclusive) satisfying the following condition, modulo \(10^9 + 7\):

  • The digits \(a_i\) for all \(i\ (1 \le i \le N)\) show up at least once in the integer.

Input Specification

The first line will contain the integer \(N\ (1 \le N \le 10)\).

The next line will contain \(N\) integers, \(a_i\ (0 \le a_i \le 9)\). It is guaranteed \(a_i\) is distinct.

The next line will contain the integer \(K\ (0 \le K < 10^{1000})\).

Output Specification

Output the number of integers satisfying the condition, modulo \(10^9+7\).

Sample Input 1

1
1
11

Sample Output 1

3

Explanation For Sample 1

The three integers satisfying the constraints are:

  • 1
  • 10
  • 11

Sample Input 2

5
3 9 2 0 1
512309002821093

Sample Output 2

891474356

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