Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1
    kitai = [[[0]*(c[1]+c[2]+c[3]+2) for _ in range(c[2]+c[3]+2)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(c[2]+c[3]+1):
            for i in range(c[1]+c[2]+c[3]+1):
                tot = i+j+k
                if tot == 0:
                    continue
                kitai[k][j][i] = N + kitai[k][j][i-1]*i + kitai[k][j-1][i+1]*j
                kitai[k][j][i] += kitai[k-1][j+1][i]*k
                kitai[k][j][i] /= tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Runtime Error):
#!/usr/bin/env python3
import sys
from collections import Counter


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[0]*(c[1]+c[2]+c[3]+1) for _ in range(c[2]+c[3]+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(c[2]+c[3]+1):
            for i in range(c[1]+c[2]+c[3]+1):
                tot = i+j+k
                if tot == 0:
                    continue
                kitai[k][j][i] = N + kitai[k][j][i-1]*i + kitai[k][j-1][i+1]*j \
                    + kitai[k-1][j+1][i]*k
                kitai[k][j][i] /= tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[0]*(N+2) for _ in range(N+2)] for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(c[2]+c[3]+1):
            for i in range(c[1]+c[2]+c[3]+1):
                tot = i+j+k
                if tot == 0:
                    continue
                res = N
                res += kitai[k][j][i-1]*i
                res += kitai[k][j-1][i+1]*j
                res += kitai[k-1][j+1][i]*k
                res /= tot
                kitai[k][j][i] = res
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[0]*(N+2) for _ in range(N+2)] for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(c[2]+c[3]+1):
            for i in range(c[1]+c[2]+c[3]+1):
                if i == 0 and j == 0 and k == 0:
                    continue
                res = N
                if i:
                    res += kitai[k][j][i-1]*i
                if j:
                    res += kitai[k][j-1][i+1]*j
                if k:
                    res += kitai[k-1][j+1][i]*k
                res /= (i+j+k)
                kitai[k][j][i] = res
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[0]*(N+2) for _ in range(N+2)] for _ in range(c[3]+1)]
    kitai[0][0][0] = 0
    for k in range(c[3]+1):
        for j in range(N+1):
            for i in range(N+1):
                if i == 0 and j == 0 and k == 0:
                    continue
                res = N
                if i:
                    res += kitai[k][j][i-1]*i
                if j:
                    res += kitai[k][j-1][i+1]*j
                if k:
                    res += kitai[k-1][j+1][i]*k
                res /= (i+j+k)
                kitai[k][j][i] = res
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter
INF = float("inf")


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[0]*(N+2) for _ in range(N+2)] for _ in range(N+2)]
    kitai[0][0][0] = 0
    for k in range(c[3]+1):
        for j in range(N+1):
            for i in range(N+1):
                if i == 0 and j == 0 and k == 0:
                    continue
                res = N
                if i - 1 >= 0:
                    res += kitai[i-1][j][k]*i
                if j - 1 >= 0:
                    res += kitai[i+1][j-1][k]*j
                if k - 1 >= 0:
                    res += kitai[i][j+1][k-1]*k
                res /= (i+j+k)
                kitai[i][j][k] = res
    print(kitai[c[1]][c[2]][c[3]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter
INF = float("inf")


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[0]*(N+2) for _ in range(N+2)] for _ in range(N+2)]
    kitai[0][0][0] = 0
    for k in range(N+1):
        for j in range(N+1):
            for i in range(N+1):
                if i == 0 and j == 0 and k == 0:
                    continue
                res = N
                if i - 1 >= 0:
                    res += kitai[i-1][j][k]*i
                if j - 1 >= 0:
                    res += kitai[i+1][j-1][k]*j
                if k - 1 >= 0:
                    res += kitai[i][j+1][k-1]*k
                res /= (i+j+k)
                kitai[i][j][k] = res
    print(kitai[c[1]][c[2]][c[3]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
#!/usr/bin/env python3
import sys
from collections import Counter
import functools
sys.setrecursionlimit(10**8)
INF = float("inf")


def solve(N: int, a: "List[int]"):

    c = Counter(a)
    kitai = [[[-1]*310 for _ in range(310)] for _ in range(310)]

    @functools.lru_cache(maxsize=None)
    def rec(i, j, k):
        if kitai[i][j][k] >= 0:
            return kitai[i][j][k]
        if i == 0 and j == 0 and k == 0:
            return 0

        res = 0
        if i - 1 >= 0:
            res += rec(i-1, j, k)*i
        if j - 1 >= 0:
            res += rec(i+1, j-1, k)*j
        if k - 1 >= 0:
            res += rec(i, j+1, k-1)*k
        res += N
        res /= (i+j+k)
        kitai[i][j][k] = res
        return res

    print(rec(c[1], c[2], c[3]))

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()


Fixed submission:
#!/usr/bin/env python3
import sys


def solve(N: int, a: "List[int]"):

    c = [0]*4
    for A in a:
        c[A] += 1

    J = min(c[2]+c[3]+1, N+1)
    I = min(c[1]+c[2]+c[3]+1, N+1)
    kitai = [[[0]*(I+1) for _ in range(J+1)]
             for _ in range(c[3]+1)]
    for k in range(c[3]+1):
        for j in range(J):
            for i in range(I):
                tot = i+j+k
                if tot == 0 or tot > N:
                    continue
                res = N + kitai[k][j][i-1]*i + \
                    kitai[k][j-1][i+1]*j + kitai[k-1][j+1][i]*k
                kitai[k][j][i] = res/tot
    print(kitai[c[3]][c[2]][c[1]])

    return


def main():

    def iterate_tokens():
        for line in sys.stdin:
            for word in line.split():
                yield word
    tokens = iterate_tokens()
    N = int(next(tokens))  # type: int
    a = [int(next(tokens)) for _ in range(N)]  # type: "List[int]"
    solve(N, a)


if __name__ == '__main__':
    main()