Lucky Draw

Time limit2sMemory limit512 MB

Summary

For n players with k lives each flipping a biased coin every round, compute the probability the game ends in a draw.

Level: Medium7 of 10
Topics: Dynamic programming, Probability, Combinatorics, Math
Solved: No attempts yet

Problem

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 the probability is 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 ≤ n ≤ 50, the number of players, 1 ≤ k ≤ 50, the number of lives each player has, and a floating point number 0.1 ≤ p ≤ 0.9, the probability the coin lands heads.

Output

Output a single floating point number: the probability of the game ending in a draw. Your answer should have an absolute error of at most 10−6.

Examples3

Example 1
Input
```
2 2 0.5
```
Expected output
```
0.185185185
```
Example 2
Input
```
2 2 0.8
```
Expected output
```
0.056241426
```
Example 3
Input
```
5 3 0.85
```
Expected output
```
0.045463964
```