F - Arcs on a Circle

Time Limit: 5 sec / Memory Limit: 512 MB

### 問題文

それぞれの円弧 i は、円周上の一様ランダムな位置に配置されます。 すなわち、円周上のランダムな点が選ばれ、その点を中心とした長さ L_i の円弧が出現します。

これらの円弧は、それぞれ独立に配置されます。例えば、円弧が交差したり、ある円弧が別の円弧を含むことがあります。

### 制約

• 2 \leq N \leq 6
• 2 \leq C \leq 50
• 1 \leq L_i < C
• 入力値はすべて整数である。

### 入力

N C
L_1 L_2 ... L_N


### 入力例 1

2 3
2 2


### 出力例 1

0.3333333333333333


### 入力例 2

4 10
1 2 3 4


### 出力例 2

0.0000000000000000


### 入力例 3

4 2
1 1 1 1


### 出力例 3

0.5000000000000000


### 入力例 4

3 5
2 2 4


### 出力例 4

0.4000000000000000


### 入力例 5

4 6
4 1 3 2


### 出力例 5

0.3148148148148148


### 入力例 6

6 49
22 13 27 8 2 19


### 出力例 6

0.2832340720702695


Score : 2100 points

### Problem Statement

You have a circle of length C, and you are placing N arcs on it. Arc i has length L_i.

Every arc i is placed on the circle uniformly at random: a random real point on the circle is chosen, then an arc of length L_i centered at this point appears.

Note that the arcs are placed independently. For example, they may intersect or contain each other.

What is the probability that every real point of the circle will be covered by at least one arc? Assume that an arc covers its ends.

### Constraints

• 2 \leq N \leq 6
• 2 \leq C \leq 50
• 1 \leq L_i < C
• All input values are integers.

### Input

Input is given from Standard Input in the following format:

N C
L_1 L_2 ... L_N


### Output

Print the probability that every real point of the circle will be covered by at least one arc. Your answer will be considered correct if its absolute error doesn't exceed 10^{-11}.

### Sample Input 1

2 3
2 2


### Sample Output 1

0.3333333333333333


The centers of the two arcs must be at distance at least 1. The probability of this on a circle of length 3 is 1 / 3.

### Sample Input 2

4 10
1 2 3 4


### Sample Output 2

0.0000000000000000


Even though the total length of the arcs is exactly C and it's possible that every real point of the circle is covered by at least one arc, the probability of this event is 0.

### Sample Input 3

4 2
1 1 1 1


### Sample Output 3

0.5000000000000000


### Sample Input 4

3 5
2 2 4


### Sample Output 4

0.4000000000000000


### Sample Input 5

4 6
4 1 3 2


### Sample Output 5

0.3148148148148148


### Sample Input 6

6 49
22 13 27 8 2 19


### Sample Output 6

0.2832340720702695