## A Counting Problem 2

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

Author:
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