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D - Poker /

Time Limit: 2 sec / Memory Limit: 1024 MB

### 問題文

1, 2, \dots, 9 が表に書かれたカードが K 枚ずつ、計 9K 枚のカードがあります。
これらのカードをランダムにシャッフルして、高橋くんと青木くんにそれぞれ、4 枚を表向きに、1 枚を裏向きにして配りました。

S, T5 文字の文字列で、先頭 4 文字は 1, 2, \dots, 9 からなり、表向きのカードに書かれた数を表します。 末尾 1 文字は # であり、裏向きのカードであることを表します。
5 枚の手札の点数を、c_i をその手札に含まれる i の枚数として、\displaystyle \sum_{i=1}^9 i \times 10^{c_i} で定義します。

### 制約

• 2 ≤ K ≤ 10^5
• |S| = |T| = 5
• S, T1 文字目から 4 文字目は 1, 2, \dots, 9 のいずれか
• 1, 2, \dots, 9 はそれぞれ、ST に合計 K 回までしか出現しない
• S, T5 文字目は #

### 入力

K
S
T


### 入力例 1

2
1144#
2233#


### 出力例 1

0.4444444444444444


### 入力例 2

2
9988#
1122#


### 出力例 2

1.0


### 入力例 3

6
1122#
2228#


### 出力例 3

0.001932367149758454


### 入力例 4

100000
3226#
3597#


### 出力例 4

0.6296297942426154


Score : 400 points

### Problem Statement

We have 9K cards. For each i = 1, 2, \dots, 9, there are K cards with i written on it.
We randomly shuffled these cards and handed out five cards - four face up and one face down - to each of Takahashi and Aoki.
You are given a string S representing the cards handed out to Takahashi and a string T representing the cards handed out to Aoki.
S and T are strings of five characters each. Each of the first four characters of each string is 1, 2, \dots, or 9, representing the number written on the face-up card. The last character of each string is #, representing that the card is face down.
Let us define the score of a five-card hand as \displaystyle \sum_{i=1}^9 i \times 10^{c_i}, where c_i is the number of cards with i written on them.
Takahashi wins when the score of Takahashi's hand is higher than that of Aoki's hand.
Find the probability that Takahashi wins.

### Constraints

• 2 ≤ K ≤ 10^5
• |S| = |T| = 5
• The first through fourth characters of each of S and T are 1, 2, \dots, or 9.
• Each of the digit 1, 2, \dots, and 9 appears at most K times in total in S and T.
• The fifth character of each of S and T is #.

### Input

Input is given from Standard Input in the following format:

K
S
T


### Output

Print the probability that Takahashi wins, as a decimal. Your answer will be judged as correct when its absolute or relative error from our answer is at most 10^{-5}.

### Sample Input 1

2
1144#
2233#


### Sample Output 1

0.4444444444444444


For example, if Takahashi's hand is 11449 and Aoki's hand is 22338, Takahashi's score is 100+2+3+400+5+6+7+8+90=621 and Aoki's score is 1+200+300+4+5+6+7+80+9=612, resulting in Takahashi's win.
Takahashi wins when the number on his face-down card is greater than that of Aoki's face-down card, so Takahashi will win with probability \frac49.

### Sample Input 2

2
9988#
1122#


### Sample Output 2

1.0


### Sample Input 3

6
1122#
2228#


### Sample Output 3

0.001932367149758454


Takahashi wins only when Takahashi's hand is 11222 and Aoki's hand is 22281, with probability \frac2{1035}.

### Sample Input 4

100000
3226#
3597#


### Sample Output 4

0.6296297942426154