2016/round2/carnival_coins.md

At the local carnival you find a midway game advertising "TONS OF FABULOUS
PRIZES". Certainly a fabulous prize or two would make your time worthwhile. It
turns out that "TONS" is actually a bit of of an understatement. There are in
fact _infinitely_ many prizes available. Consequently, the game operator is
willing to give you a chance to score multiple prizes in a single game (for a
nominal fee).

After you hand over your money, the game operator gives you **N** coins. Each
coin has the same probability **p** of landing on heads when flipped (and
consequently probability 1 - **p** of landing on tails). She also gives you a
goal, an integer **K**.

As long as you still have some coins remaining, you can select any number of
them and flip them all simultaneously. These coins are then taken away. If at
least **K** of them land on heads, you win a prize. If you play optimally,
what is the expected number of prizes you'll manage to win?

### Input

Input begins with an integer **T**, the number of times you play the game. For
each attempt, there is a line containing the space-separated values **N**,
**K** and **p**. **N** and **K** are integers, and **p** is given with at most
16 decimal places.

### Output

For the **i**th attempt, print a line containing "Case #**i**: " followed by
the expected number of prizes you'll win. Your output should have at most 10-6
absolute or relative error.

### Constraints

1 ≤ **T** ≤ 100  
1 ≤ **N** ≤ 3,000  
1 ≤ **K** ≤ 3,000  
0 ≤ **p** ≤ 1  

### Explanation of Sample

In the first case, flipping the coins together gives you a 75% chance of
winning a prize. If you flip them separately, you get a 50% chance on each
flip. The latter approach is better, giving you 1 expected prize rather than
0.75 expected prizes. In the second case, it is optimal to partition the ten
coins into two sets of five. In the third case, it is optimal to flip all ten
coins at once.