Andrew and the Birthday

Today is Andrew's birthday.
He has invited N friends for celebrating it, and now he needs to prepare the table for them. All his friends will be sitting in a row on the seats numbered from 1 to N, inclusive. There are M different types of meatballs (numbered from 1 to M, inclusive). Andrew is going to serve meatballs of exactly one type at each of the seats.
You are an given array A of N integers in range [0, M]. A_{i} = 0 means that Andrew is not sure which type of meatballs he should serve at seat i. So, in this case, we consider seat i will be served meatballs of types from 1 to M with equal probability. Otherwise, A_{i} ≥ 1 means that seat i will be served meatballs of type A_{i}.
It's very important for Andrew to know, what is the expected maximal number of consecutively sitting friends with the same type of meatballs served at their seats. Help Andrew to find this number.
Input
The first line of the input contains an integer T, denoting the number of test cases. The description of T test cases follows. The first line of each test case contains two spaceseparated integers N and M. The second line contains N spaceseparated integers A_{1}, A_{2}, ..., A_{N}.
Output
For each test case, output a real number, denoting the expected value of maximal number of consecutively sitting friends with the same type of meatballs served to their seats.
The output values will be considered as correct if and only if they have the absolute errors at most 10^{−6}.
Constraints
 1 ≤ T ≤ 10
 1 ≤ N, M ≤ 50
 0 ≤ A_{i} ≤ M
Example
Input: 2 2 2 0 0 4 7 0 1 1 0 Output: 1.500000000 2.285714286
Explanation
Example case 1: There are 4 possibilities (with equal probability) of the serving as follows.
1 1 : maximal number of consecutive seats with the same type = 2
1 2 : maximal number of consecutive seats with the same type = 1
2 1 : maximal number of consecutive seats with the same type = 1
2 2 : maximal number of consecutive seats with the same type = 2
Therefore the answer is 2 * (1/4) + 1 * (1/4) + 1 * (1/4) + 2 * (1/4) = 1.5.
Example case 2: There are 49 possibilities of the serving. Among these, the patterns with maximal number of consecutive seats with the same type = 3 are the following 12 patterns
1 1 1 x, where x = 2, 3, ..., 7
x 1 1 1, where x = 2, 3, ..., 7
and the pattern with maximal number of consecutive seats with the same type = 4 is the following 1 pattern
1 1 1 1
In other 36 patterns, maximal number of consecutive seats with the same type = 2. Therefore the answer is 2 * (36/49) + 3 * (12/49) + 4 * (1/49) ≈ 2.2857.
Author:  witua 
Tester:  laycurse 
Editorial  http://discuss.codechef.com/problems/LEFEED 
Tags  cook33, medium, probability, witua 
Date Added:  20032012 
Time Limit:  1 sec 
Source Limit:  50000 Bytes 
Languages:  C, CPP14, JAVA, PYTH, PYTH 3.6, CS2, PAS fpc, PAS gpc, RUBY, PHP, GO, NODEJS, HASK, SCALA, D, PERL, FORT, WSPC, ADA, CAML, ICK, BF, ASM, CLPS, PRLG, ICON, SCM qobi, PIKE, ST, NICE, LUA, BASH, NEM, LISP sbcl, LISP clisp, SCM guile, JS, ERL, TCL, PERL6, TEXT, CLOJ, FS 
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