| Let **S** be a sequence of **N** natural numbers. We can define an infinite |
| sequence **MS** in the following way: **MS**[k] = **S**[k mod **N**] + **N** * |
| floor(k / **N**). Where k is a zero based index. |
|
|
| For example if the sequence **S** is {2, 1, 3} then **MS** would be {2, 1, 3, |
| 5, 4, 6, 8, 7, 9, 11, 10, 12...} |
|
|
| Now consider a subsequence of **MS** generated by picking two random indices |
| **a**, **b** from the range [**0**..**R**] inclusive, and taking all the |
| elements between them, that is: **MS**[min(**a**, **b**)..max(**a**, **b**)]. |
|
|
| If we use the same **MS** as in the example above and **a** = 2, **b** = 5 |
| then our subsequence would be {3, 5, 4, 6}. |
|
|
| Your task is to calculate the probability that the selected subsequence has at |
| least **K** distinct elements. **a** and **b** are selected independently and |
| with a uniform distribution. The result should be printed as a fraction. See |
| the "Output" section for clarification. |
|
|
| ### Input |
|
|
| The first line of the input file contains an integer **T**. This is followed |
| by **T** test cases, each of which has two lines. |
|
|
| The first line of each test case contains three integers separated by spaces, |
| **N**, **K**, and **R**. |
|
|
| The second line contains **N** space separated integers, **S**[0] through |
| **S**[**N**-1]. |
|
|
| ### Constraints |
|
|
| 1 ≤ **T** ≤ 20 |
| 1 ≤ **N** ≤ 2,000 |
| 1 ≤ **K** ≤ **R** ≤ 1,000,000,000 |
| 1 ≤ **S**[i] ≤ 100,000 |
|
|
| ### Output |
|
|
| For each of the test cases numbered in order from 1 to **T**, output "Case |
| #**i**: " followed by the probability that the selected subsequence of **MS** |
| has at least **K** distinct elements. The probability should be expressed as a |
| fraction **p**/**q**, where **p** and **q** represent the numerator and |
| denominator respectively and are relatively prime (that is they share no |
| common positive divisors except 1). |
|
|
| If the probability is 0 or 1 output 0/1 or 1/1 respectively. |
|
|
| ### Examples |
|
|
| In the first example there are 36 different subsequences to consider. 6 of |
| them have only a single number, and the remaining 30 have at least 2 different |
| numbers, so the answer is 5/6. |
|
|
| The second example is similar, but now the sequence looks like {2, 1, 5, 5, 4, |
| 8}. There are 8 subsequences with less than 2 distinct numbers: the six single |
| number subsequences plus (a=2, b=3) and (a=3, b=2) which both result in {5,5}. |
| That gives a probability of (36 - 8) / 36 = 7/9. |
|
|
| The third example uses the same sequence as the second example, but now we |
| want to have subsequences with at least 4 different numbers. All pairs of |
| indices that have this property are: (0,4), (0, 5), (1, 5), (4, 0), (5, 0), |
| and (5, 1). Six out of thirty six results in a probability of 1/6. |
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