The Last Carrot

Time: 1.0 s     Memory: 1024 MB
  • 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
    
Chalmers Coding Camp 24 problems
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  • A The Last Carrot
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