# Tribbles

GRAVITATION, n. “The tendency of all bodies to approach one another with a strength proportion to the quantity of matter they contain – the quantity of matter they contain being ascertained by the strength of their tendency to approach one another. This is a lovely and edifying illustration of how science, having made A the proof of B, makes B the proof of A.” Ambrose Bierce You have a population of k Tribbles. This particular species of Tribbles live for exactly one day and then die. Just before death, a single Tribble has the probability Pi of giving birth to i more Tribbles. What is the probability that after m generations, every Tribble will be dead? Input The first line of input gives the number of cases, N. N test cases follow. Each one starts with a line containingn(1≤n≤1000),k(0≤k≤1000)andm(0≤m≤1000). Thenextnlineswillgivethe probabilities P0, P1, . . . , Pn−1. Output For each test case, output one line containing ‘Case #x:’ followed by the answer, correct up to an absolute or relative error of 10−6. Sample Input 4 311 0.33 0.34 0.33 312 0.33 0.34 0.33 312 0.5 0.0 0.5 422 0.5 0.0 0.0 0.5

2/2 Sample Output Case #1: 0.3300000 Case #2: 0.4781370 Case #3: 0.6250000 Case #4: 0.3164062