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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