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yeetzhee.py
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yeetzhee.py
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# Copyright (c) 2020 kamyu. All rights reserved.
#
# Google Kick Start 2020 Round F - Problem D. Yeetzhee
# https://codingcompetitions.withgoogle.com/kickstart/round/000000000019ff48/00000000003f4dea
#
# Time: O(M * S), S is the number of states with sum <= M and length <= K, which is at most sum(partition(i) for i in [0..M])
# Space: O(M * S)
#
# if M = 50, the number of states is at most 1,295,971
# see https://oeis.org/A000041/list,
# p[0..50] = [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173525, 204226]
#
def memoization(A, curr, lookup): # O(M) time for each state
if tuple(curr) not in lookup:
count, total = 1.0, 0.0
left = 0
while left < len(curr):
right = left
while right+1 < len(curr) and curr[right+1] == curr[left]:
right += 1
if curr[right]+1 <= A[right]: # greedily fill and keep curr sorted
curr[right] += 1
count += memoization(A, curr, lookup)*(right-left+1)/len(A)
curr[right] -= 1
total += float(right-left+1)/len(A)
left = right+1
lookup[tuple(curr)] = count/total # ev = (1+sum(pi*ei)) / sum(pi)
return lookup[tuple(curr)]
def yeetzhee():
N, M, K = map(int, raw_input().strip().split())
A = [0]*(M-K)
for _ in xrange(K):
A.append(input())
return memoization(A, [0]*M, {tuple(A):0.0})
for case in xrange(input()):
print 'Case #%d: %s' % (case+1, yeetzhee())