### Problem Statement

Taro likes apples very much. He has N boxes numbered from 0 to N-1. There are K different types of apples numbered from 0 to K-1. You are given three String[]s, hundred, ten and one. Concatenate j-th characters in hundred[i], ten[i] and one[i] in this order to get a string that represents the number of j-th type apples in Box i (it may have leading zero(s)). This number will be between 0 and 199, inclusive.

He decided to choose one apple from his boxes, and he does so in the following way:

• First Step: He chooses a non-empty subset of his N boxes randomly and transfers all apples from those boxes to another box (this is a box other than the original N boxes and it is initially empty). Each non-empty subset of boxes has the same probability of being chosen.

• Second Step: He chooses one apple from the new box randomly. Each apple in the box has the same probability of being chosen.

Return a double[] that contains exactly K elements and whose i-th element is the probability that Taro chooses an i-th type apple.

### Definition

 Class: RandomApple Method: theProbability Parameters: String[], String[], String[] Returns: double[] Method signature: double[] theProbability(String[] hundred, String[] ten, String[] one) (be sure your method is public)

### Notes

-Your return value must have an absolute or relative error less than 1e-9.

### Constraints

-N will be between 1 and 50, where N is the number of elements in hundred.
-K will be between 1 and 50, where K is the number of characters in hundred.
-ten and one will contain exactly N elements.
-Each element in hundred, ten and one will contain exactly K characters.
-Each character in hundred will be '0' or '1'.
-Each character in ten and one will be a digit ('0'-'9').
-Each box will contain at least one apple.

### Examples

0)

 `{"00"}` `{"00"}` `{"58"}`
`Returns: {0.38461538461538464, 0.6153846153846154 }`
 There is only one box which contains 5 type-0 apples and 8 type-1 apples. The probability of choosing a type-0 apple is 5 / 13.
1)

 `{"00", "00"}` `{"00", "00"}` `{"21", "11"}`
`Returns: {0.5888888888888889, 0.4111111111111111 }`
 If he chooses only box 0 in the first step, the probability of choosing a type-0 apple is 2 / 3. If he chooses only box 1 in the first step, the probability of choosing a type-0 apple is 1 / 2. If he chooses both boxes in the first step, the probability of choosing a type-0 apple is 3 / 5. So the probability of choosing a type-0 apple is (2 / 3 + 1 / 2 + 3 / 5) / 3 = 53 / 90.
2)

 `{"0000", "0000", "0000"}` `{"2284", "0966", "9334"}` `{"1090", "3942", "4336"}`
```Returns:
{0.19685958571981937, 0.24397246802233483, 0.31496640865458775, 0.24420153760325805 }```
3)

 `{"01010110", "00011000", "00001000", "10001010", "10111110"}` `{"22218214", "32244284", "68402430", "18140323", "29043145"}` `{"87688689", "36101317", "69474068", "29337374", "87255881"}`
```Returns:
{0.11930766223754977, 0.14033271060661345, 0.0652282589028571, 0.14448118133046356, 0.1981894622733832, 0.10743462836879789, 0.16411823601857622, 0.06090786026175882 }```
4)

 `{"10"}` `{"00"}` `{"00"}`
`Returns: {1.0, 0.0 }`
 One box with 100 type-0 apples and no type-1 apples.

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=11026

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=14187&pm=11026

rng_58

#### Testers:

liympanda , konqueror , ivan_metelsky , keshav_57

#### Problem categories:

Dynamic Programming, Math