Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
import sys
input = sys.stdin.buffer.readline
from itertools import product

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dic = {}
    for p in product(range(n+1),repeat=3):
        dic['{}-{}-{}'.format(*p)] = .0
    n = float(n)
    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        key = '{}-{}-{}'.format(i,j,k)
        if key in dic and dic[key] > 0:
            return dic[key]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        dic[key] = ans
        return ans

    ans = exp(a[1],a[2],a[3])
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    n = float(n)
    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        if dp[i][j][k]:
            return dp[i][j][k]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        dp[i][j][k] = ans
        return ans

    ans = exp(a[1],a[2],a[3])
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
def main():
    from collections import Counter
    import sys
    input = sys.stdin.buffer.readline
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i+j+k == 0:
                    continue
                else:
                    tmp = n
                    if i > 0:
                        tmp += i * dp[i-1][j][k]
                    if j > 0:
                        tmp += j * dp[i+1][j-1][k]
                    if k > 0:
                        tmp += k * dp[i][j+1][k-1]
                    dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])
if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dic = {}

    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        key = '{}-{}-{}'.format(i,j,k)
        if key in dic:
            return dic[key]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        dic[key] = ans
        return ans

    ans = exp(a[1],a[2],a[3])
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
from collections import Counter
def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[''] * (310) for _ in range(310)] for _ in range(310)]
    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        if dp[i][j][k]:
            return dp[i][j][k]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        dp[i][j][k] = ans
        return ans

    ans = exp(a[1],a[2],a[3])
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
def main():
    n = int(input())
    a = list(map(int, input().split()))
    dp = [[[''] * (310) for _ in range(310)] for _ in range(310)]
    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        if dp[i][j][k]:
            return dp[i][j][k]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        dp[i][j][k] = ans
        return ans

    ans = exp(a.count(1),a.count(2),a.count(3))
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
def main():
    n = int(input())
    a = list(map(int, input().split()))
    dic = {}

    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        key = '{}-{}-{}'.format(i,j,k)
        if key in dic:
            return dic[key]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        dic[key] = ans
        return ans

    ans = exp(a.count(1),a.count(2),a.count(3))
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()

Buggy python submission (verdict: Time Limit Exceeded):
def main():
    n = int(input())
    a = list(map(int, input().split()))
    dic = {}

    def exp(i,j,k):
        if i + j + k == 0:
            return 0
        key = '{}-{}-{}'.format(i,j,k)
        if key in dic:
            return dic[key]
        ans = n
        if i > 0:
            ans += i * exp(i-1,j,k)
        if j > 0:
            ans += j * exp(i+1,j-1,k)
        if k > 0:
            ans += k * exp(i,j+1,k-1)
        ans /= (i+j+k)
        return ans

    ans = exp(a.count(1),a.count(2),a.count(3))
    print(ans)

if __name__ == '__main__':
    main()

Fixed submission:
from collections import Counter
import sys
input = sys.stdin.buffer.readline

def main():
    n = int(input())
    a = Counter(map(int, input().split()))
    dp = [[[.0] * (n+1) for _ in range(n+1)] for _ in range(n+1)]
    for k in range(a[3]+1):
        for j in range(a[2]+a[3]-k+1):
            for i in range(n-j-k+1):
                if i + j + k == 0:
                    continue
                tmp = n
                if i > 0:
                    tmp += i * dp[i-1][j][k]
                if j > 0:
                    tmp += j * dp[i+1][j-1][k]
                if k > 0:
                    tmp += k * dp[i][j+1][k-1]
                dp[i][j][k] = tmp / (i+j+k)
    print(dp[a[1]][a[2]][a[3]])

if __name__ == '__main__':
    main()