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It’s frustrating for Tor and his mother that sometimes there’s a carrot left (see the task Carrots). Instead of sharing it, Tor suggests rolling dice to decide who gets the carrot. He takes out $N$ dice, each with $M$ sides, numbered from $1$ to $M$, each with an equal probability of being rolled. He then lets his mother choose $K$ numbers, and if the numbers on the dice sum up to one of those numbers, she wins; otherwise, Tor wins. Now, Mother wants your help to write a program to determine her chances of winning if she chooses her outcomes optimally.
Input
Input consists of the the three integers $N, M, K$ ($1 \le N \le 20$, $1 \le M \le 5000$, $1 \le K < N \cdot M$), the number of dice, how many sides each dice has and how many outcomes Mother can choose.
Output
Print a real number – the probability that Mother will win if she chooses her outcomes optimally. The answer will be considered correct if it has an absolute error of at most $10^{-5}$.
Scoring
Your solution will be tested on a set of test groups, each worth a number of points. Each test group contains a set of test cases. To get the points for a test group you need to solve all test cases in the test group.
Group
Point value
Constraints
$1$
$20$
$n=2,m \le 100 $
$2$
$20$
$n \le 6, m \le 6 $
$3$
$20$
$n \le 12, m \le 12 $
$4$
$20$
$n \le 20, m \le 100 $
$5$
$20$
$n \le 20, m \le 5000 $
Sample Input 1 Sample Output 1 2 6 2
0.305555555555556
Sample Input 2 Sample Output 2 3 7 4
0.41399416909621
Sample Input 3 Sample Output 3 20 2189 2734
0.369028440650235
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To solve the problems, you can either start a virtual contest or register for regular practice. A virtual contest simulates a participation in the original contest with a duration of 4 days, while regular practice lets you submit solutions without any constraints.
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