Kattis

# Lucky Draw

You and your friends at the Betting against All Probability Club are visiting a casino where the following game is played.

Each of the $n$ players starts with $k$ lives and puts in a fixed amount of money. In each round of the game, each player flips a biased coin and loses a life if she gets tails. The game ends when only one player remains, in which case this person wins, or ends in a draw if all remaining players lose their last life in the same round. If there is a winner, she wins $n$ times her original bet. In case of a draw, no one wins anything.

Being a BAPC member you quickly realize the casino has an edge here: whenever the game ends in a draw all of the contestants lose the money they bet. You are now wondering what exactly is the probability that this game ends in a draw, so you can figure out how much the casino profits on average.

## Input

• One line containing two integers, $2\leq n\leq 50$, the number of players, $1\leq k\leq 50$, the number of lives each player has, and a real number $0.1 \leq p \leq 0.9$, with at most two digits after the decimal point, which is the probability the coin lands heads.

## Output

• Output a single real number: the probability of the game ending in a draw. Your answer should have an absolute error of at most $10^{-6}$.

Sample Input 1 Sample Output 1
2 2 0.5
0.185185185
Sample Input 2 Sample Output 2
2 2 0.8
0.056241426
Sample Input 3 Sample Output 3
5 3 0.85
0.045463964