Problem:


Score : 100 points

Problem Statement

There are N dishes, numbered 1, 2, \ldots, N.
Initially, for each i (1 \leq i \leq N), Dish i has a_i (1 \leq a_i \leq 3) pieces of sushi on it.

Taro will perform the following operation repeatedly until all the pieces of sushi are eaten:

Roll a die that shows the numbers 1, 2, \ldots, N with equal probabilities, and let i be the outcome. If there are some pieces of sushi on Dish i, eat one of them; if there is none, do nothing.

Find the expected number of times the operation is performed before all the pieces of sushi are eaten.

Constraints

All values in input are integers.

1 \leq N \leq 300

1 \leq a_i \leq 3

Input

Input is given from Standard Input in the following format:

N
a_1 a_2 \ldots a_N

Output

Print the expected number of times the operation is performed before all the pieces of sushi are eaten.
The output is considered correct when the relative difference is not greater than 10^{-9}.

Sample Input 1

3
1 1 1

Sample Output 1

5.5

The expected number of operations before the first piece of sushi is eaten, is 1.
After that, the expected number of operations before the second sushi is eaten, is 1.5.
After that, the expected number of operations before the third sushi is eaten, is 3.
Thus, the expected total number of operations is 1 + 1.5 + 3 = 5.5.

Sample Input 2

1
3

Sample Output 2

3

Outputs such as 3.00, 3.000000003 and 2.999999997 will also be accepted.

Sample Input 3

2
1 2

Sample Output 3

4.5

Sample Input 4

10
1 3 2 3 3 2 3 2 1 3

Sample Output 4

54.48064457488221



Sample input:
3
1 1 1


Sample output:
5.5

Submission status: Time Limit Exceeded

Solution code:
import sys
input=sys.stdin.readline
n=int(input())
cnt=[0]*3
for i in map(int,input().split()):
    cnt[i-1]+=1
c1,c2,c3=cnt
DP=[[[0]*(n+1) for _ in range(n+1)] for _ in range(n+1)] 
DP[0][0][0]=0
for i in range(c3+1):
    for j in range(c3+c2-i+1):
        for k in range(n-i-j+1):
            if i==j==k==0:
                continue
            tmp=n
            if i:
                tmp+=DP[k][j+1][i-1]*i
            if j:
                tmp+=DP[k+1][j-1][i]*j
            if k:
                tmp+=DP[k-1][j][i]*k
            DP[k][j][i]=tmp/(i+j+k)
print(DP[c1][c2][c3])

Submission status: Time Limit Exceeded

Solution code:
import sys
input=sys.stdin.readline
from collections import Counter
n=int(input())
A=tuple(map(int,input().split()))
C=Counter(A)
c1,c2,c3=[C[i] for i in (1,2,3)]
DP=[[[0]*(n+1) for _ in range(n+1)] for _ in range(n+1)] 
DP[0][0][0]=0
for i in range(c3+1):
    for j in range(c3+c2-i+1):
        for k in range(n-i-j+1):
            if i==j==k==0:
                continue
            tmp=n
            if i:
                tmp+=DP[k][j+1][i-1]*i
            if j:
                tmp+=DP[k+1][j-1][i]*j
            if k:
                tmp+=DP[k-1][j][i]*k
            DP[k][j][i]=tmp/(i+j+k)
print(DP[c1][c2][c3])

Submission status: Time Limit Exceeded

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = SR()
    return l
mod = 1000000007
sys.setrecursionlimit(1000000)

#A
def A():
    n = I()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n-2):
        dp[i+1] = min(dp[i+1],dp[i]+abs(h[i+1]-h[i]))
        dp[i+2] = min(dp[i+2],dp[i]+abs(h[i+2]-h[i]))
    dp[n-1] = min(dp[n-1],dp[n-2]+abs(h[n-1]-h[n-2]))
    print(dp[n-1])
#B
def B():
    n,k = LI()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n):
        for j in range(1,min(n-i,k+1)):
            dp[i+j] = min(dp[i+j],dp[i]+abs(h[i+j]-h[i]))
    print(dp[n-1])

#C
def C():
    n = I()
    v = LIR(n)
    dp = [[0,0,0] for i in range(n+1)]
    for i in range(n):
        dp[i+1][0] = max(dp[i+1][0],dp[i][1]+v[i][0],dp[i][2]+v[i][0])
        dp[i+1][1] = max(dp[i+1][1],dp[i][2]+v[i][1],dp[i][0]+v[i][1])
        dp[i+1][2] = max(dp[i+1][2],dp[i][0]+v[i][2],dp[i][1]+v[i][2])
    print(max(dp[n]))

#D
def D():
    n,W = LI()
    g = LIR(n)
    dp = [[0 for i in range(W+1)] for j in range(n+1)]
    for i in range(n):
        for j in range(W+1)[::-1]:
            w,v = g[i]
            if j >= w:
                dp[i+1][j] = max(dp[i][j],dp[i][j-w]+v)
            else:
                dp[i+1][j] = max(dp[i+1][j],dp[i][j])
    print(max(dp[n]))
#E
def E():
    n,W = LI()
    g = LIR(n)
    V = 100000
    dp = [[float("inf") for i in range(V+1)] for j in range(n+1)]
    dp[0][0] = 0
    for i in range(n):
        for j in range(V+1)[::-1]:
            w,v = g[i]
            if j >= v:
                dp[i+1][j] = min(dp[i][j],dp[i][j-v]+w)
            else:
                dp[i+1][j] = min(dp[i+1][j],dp[i][j])

    for i in range(V+1)[::-1]:
        if dp[n][i] <= W:
            print(i)
            quit()

#F
def F():
    s = S()
    t = S()
    dp = [[0 for i in range(len(t)+1)] for j in range(len(s)+1)]
    for i in range(len(s)):
        for j in range(len(t)):
            if s[i] == t[j]:
                dp[i+1][j+1] = max(dp[i][j]+1,dp[i+1][j],dp[i][j+1])
            else:
                dp[i+1][j+1] = max(dp[i+1][j],dp[i][j+1])

    i = len(s)
    j = len(t)
    ans = ""
    while i > 0 and j > 0:
        if dp[i][j] == dp[i-1][j]:i -= 1
        elif dp[i][j] == dp[i][j-1]:j -= 1
        else:
            i -= 1
            j -= 1
            ans = s[i]+ans
    print(ans)
#G
def G():
    def dfs(x):
        if dp[x] != None:
            return dp[x]
        if len(v[x]) == 0:
            dp[x] = 0
            return 0
        else:
            res = 0
            for y in v[x]:
                res = max(res,dfs(y)+1)
            dp[x] = res
            return dp[x]
    n,m = LI()
    dp = [None for i in range(n)]
    v = [[] for i in range(n)]
    for i in range(m):
        x,y = LI()
        v[x-1].append(y-1)
    for i in range(n):
        dfs(i)
    print(max(dp))

#H
def H():
    h,w = LI()
    s = SR(h)
    dp = [[0 for i in range(w)] for j in range(h)]
    dp[0][0] = 1
    for i in range(h-1):
        for j in range(w-1):
            if s[i+1][j] == ".":
                dp[i+1][j] += dp[i][j]
                dp[i+1][j] %= mod
            if s[i][j+1] == ".":
                dp[i][j+1] += dp[i][j]
                dp[i][j+1] %= mod
    for j in range(w-1):
        if s[h-1][j+1] == ".":
            dp[h-1][j+1] += dp[h-1][j]
            dp[h-1][j+1] %= mod
    for i in range(h-1):
        if s[i+1][w-1] == ".":
            dp[i+1][w-1] += dp[i][w-1]
            dp[i+1][w-1] %= mod
    print(dp[h-1][w-1])

#I
def J():
    n = I()
    p = list(map(float, input().split()))
    dp = [[0 for i in range(n+1)] for j in range(n+1)]
    dp[0][0] = 1
    for i in range(n):
        for j in range(n+1):
            dp[i+1][j] = p[i]*dp[i][j-1]+(1-p[i])*dp[i][j]

    ans = 0
    m = math.floor(n/2)
    for i in range(m+1,n+1):
        ans += dp[n][i]
    print(ans)

#J
def K():
    n = I()
    a = LI()
    d1 = a.count(1)
    d2 = a.count(2)
    d3 = a.count(3)
    dp = [[[None for i in range(d3+1)] for j in range(d2+d3+1)] for k in range(n+1)]
    dp[0][0][0] = 0.0
    for k in range(d3+1):
        for j in range(d2+d3+1-k):
            for i in range(n+1-k-j):
                l = i+j+k
                if l == 0:continue
                res = n
                if i:
                    res += i*dp[i-1][j][k]
                if j:
                    res += j*dp[i+1][j-1][k]
                if k:
                    res += k*dp[i][j+1][k-1]
                dp[i][j][k] = res/l
    print(dp[d1][d2][d3])
#Solve
if __name__ == "__main__":
    K()

Submission status: Time Limit Exceeded

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = SR()
    return l
mod = 1000000007
sys.setrecursionlimit(1000000)

#A
def A():
    n = I()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n-2):
        dp[i+1] = min(dp[i+1],dp[i]+abs(h[i+1]-h[i]))
        dp[i+2] = min(dp[i+2],dp[i]+abs(h[i+2]-h[i]))
    dp[n-1] = min(dp[n-1],dp[n-2]+abs(h[n-1]-h[n-2]))
    print(dp[n-1])
#B
def B():
    n,k = LI()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n):
        for j in range(1,min(n-i,k+1)):
            dp[i+j] = min(dp[i+j],dp[i]+abs(h[i+j]-h[i]))
    print(dp[n-1])

#C
def C():
    n = I()
    v = LIR(n)
    dp = [[0,0,0] for i in range(n+1)]
    for i in range(n):
        dp[i+1][0] = max(dp[i+1][0],dp[i][1]+v[i][0],dp[i][2]+v[i][0])
        dp[i+1][1] = max(dp[i+1][1],dp[i][2]+v[i][1],dp[i][0]+v[i][1])
        dp[i+1][2] = max(dp[i+1][2],dp[i][0]+v[i][2],dp[i][1]+v[i][2])
    print(max(dp[n]))

#D
def D():
    n,W = LI()
    g = LIR(n)
    dp = [[0 for i in range(W+1)] for j in range(n+1)]
    for i in range(n):
        for j in range(W+1)[::-1]:
            w,v = g[i]
            if j >= w:
                dp[i+1][j] = max(dp[i][j],dp[i][j-w]+v)
            else:
                dp[i+1][j] = max(dp[i+1][j],dp[i][j])
    print(max(dp[n]))
#E
def E():
    n,W = LI()
    g = LIR(n)
    V = 100000
    dp = [[float("inf") for i in range(V+1)] for j in range(n+1)]
    dp[0][0] = 0
    for i in range(n):
        for j in range(V+1)[::-1]:
            w,v = g[i]
            if j >= v:
                dp[i+1][j] = min(dp[i][j],dp[i][j-v]+w)
            else:
                dp[i+1][j] = min(dp[i+1][j],dp[i][j])

    for i in range(V+1)[::-1]:
        if dp[n][i] <= W:
            print(i)
            quit()

#F
def F():
    s = S()
    t = S()
    dp = [[0 for i in range(len(t)+1)] for j in range(len(s)+1)]
    for i in range(len(s)):
        for j in range(len(t)):
            if s[i] == t[j]:
                dp[i+1][j+1] = max(dp[i][j]+1,dp[i+1][j],dp[i][j+1])
            else:
                dp[i+1][j+1] = max(dp[i+1][j],dp[i][j+1])

    i = len(s)
    j = len(t)
    ans = ""
    while i > 0 and j > 0:
        if dp[i][j] == dp[i-1][j]:i -= 1
        elif dp[i][j] == dp[i][j-1]:j -= 1
        else:
            i -= 1
            j -= 1
            ans = s[i]+ans
    print(ans)
#G
def G():
    def dfs(x):
        if dp[x] != None:
            return dp[x]
        if len(v[x]) == 0:
            dp[x] = 0
            return 0
        else:
            res = 0
            for y in v[x]:
                res = max(res,dfs(y)+1)
            dp[x] = res
            return dp[x]
    n,m = LI()
    dp = [None for i in range(n)]
    v = [[] for i in range(n)]
    for i in range(m):
        x,y = LI()
        v[x-1].append(y-1)
    for i in range(n):
        dfs(i)
    print(max(dp))

#H
def H():
    h,w = LI()
    s = SR(h)
    dp = [[0 for i in range(w)] for j in range(h)]
    dp[0][0] = 1
    for i in range(h-1):
        for j in range(w-1):
            if s[i+1][j] == ".":
                dp[i+1][j] += dp[i][j]
                dp[i+1][j] %= mod
            if s[i][j+1] == ".":
                dp[i][j+1] += dp[i][j]
                dp[i][j+1] %= mod
    for j in range(w-1):
        if s[h-1][j+1] == ".":
            dp[h-1][j+1] += dp[h-1][j]
            dp[h-1][j+1] %= mod
    for i in range(h-1):
        if s[i+1][w-1] == ".":
            dp[i+1][w-1] += dp[i][w-1]
            dp[i+1][w-1] %= mod
    print(dp[h-1][w-1])

#I
def J():
    n = I()
    p = list(map(float, input().split()))
    dp = [[0 for i in range(n+1)] for j in range(n+1)]
    dp[0][0] = 1
    for i in range(n):
        for j in range(n+1):
            dp[i+1][j] = p[i]*dp[i][j-1]+(1-p[i])*dp[i][j]

    ans = 0
    m = math.floor(n/2)
    for i in range(m+1,n+1):
        ans += dp[n][i]
    print(ans)

#J
def K():
    n = I()
    a = LI()
    d1 = a.count(1)
    d2 = a.count(2)
    d3 = a.count(3)
    dp = [[[0.0]*(d3+1) for j in range(d2+d3+1)] for k in range(n+1)]
    for k in range(d3+1):
        for j in range(d2+d3+1-k):
            for i in range(n+1-k-j):
                if i==j==k==0:continue
                l = i+j+k
                dp[i][j][k] = n
                if i:
                    dp[i][j][k] += i*dp[i-1][j][k]
                if j:
                    dp[i][j][k] += j*dp[i+1][j-1][k]
                if k:
                    dp[i][j][k] += k*dp[i][j+1][k-1]
                dp[i][j][k] /= l
    print(dp[d1][d2][d3])
#Solve
if __name__ == "__main__":
    K()

Submission status: Time Limit Exceeded

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = SR()
    return l
mod = 1000000007
sys.setrecursionlimit(1000000)

#A
def A():
    n = I()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n-2):
        dp[i+1] = min(dp[i+1],dp[i]+abs(h[i+1]-h[i]))
        dp[i+2] = min(dp[i+2],dp[i]+abs(h[i+2]-h[i]))
    dp[n-1] = min(dp[n-1],dp[n-2]+abs(h[n-1]-h[n-2]))
    print(dp[n-1])
#B
def B():
    n,k = LI()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n):
        for j in range(1,min(n-i,k+1)):
            dp[i+j] = min(dp[i+j],dp[i]+abs(h[i+j]-h[i]))
    print(dp[n-1])

#C
def C():
    n = I()
    v = LIR(n)
    dp = [[0,0,0] for i in range(n+1)]
    for i in range(n):
        dp[i+1][0] = max(dp[i+1][0],dp[i][1]+v[i][0],dp[i][2]+v[i][0])
        dp[i+1][1] = max(dp[i+1][1],dp[i][2]+v[i][1],dp[i][0]+v[i][1])
        dp[i+1][2] = max(dp[i+1][2],dp[i][0]+v[i][2],dp[i][1]+v[i][2])
    print(max(dp[n]))

#D
def D():
    n,W = LI()
    g = LIR(n)
    dp = [[0 for i in range(W+1)] for j in range(n+1)]
    for i in range(n):
        for j in range(W+1)[::-1]:
            w,v = g[i]
            if j >= w:
                dp[i+1][j] = max(dp[i][j],dp[i][j-w]+v)
            else:
                dp[i+1][j] = max(dp[i+1][j],dp[i][j])
    print(max(dp[n]))
#E
def E():
    n,W = LI()
    g = LIR(n)
    V = 100000
    dp = [[float("inf") for i in range(V+1)] for j in range(n+1)]
    dp[0][0] = 0
    for i in range(n):
        for j in range(V+1)[::-1]:
            w,v = g[i]
            if j >= v:
                dp[i+1][j] = min(dp[i][j],dp[i][j-v]+w)
            else:
                dp[i+1][j] = min(dp[i+1][j],dp[i][j])

    for i in range(V+1)[::-1]:
        if dp[n][i] <= W:
            print(i)
            quit()

#F
def F():
    s = S()
    t = S()
    dp = [[0 for i in range(len(t)+1)] for j in range(len(s)+1)]
    for i in range(len(s)):
        for j in range(len(t)):
            if s[i] == t[j]:
                dp[i+1][j+1] = max(dp[i][j]+1,dp[i+1][j],dp[i][j+1])
            else:
                dp[i+1][j+1] = max(dp[i+1][j],dp[i][j+1])

    i = len(s)
    j = len(t)
    ans = ""
    while i > 0 and j > 0:
        if dp[i][j] == dp[i-1][j]:i -= 1
        elif dp[i][j] == dp[i][j-1]:j -= 1
        else:
            i -= 1
            j -= 1
            ans = s[i]+ans
    print(ans)
#G
def G():
    def dfs(x):
        if dp[x] != None:
            return dp[x]
        if len(v[x]) == 0:
            dp[x] = 0
            return 0
        else:
            res = 0
            for y in v[x]:
                res = max(res,dfs(y)+1)
            dp[x] = res
            return dp[x]
    n,m = LI()
    dp = [None for i in range(n)]
    v = [[] for i in range(n)]
    for i in range(m):
        x,y = LI()
        v[x-1].append(y-1)
    for i in range(n):
        dfs(i)
    print(max(dp))

#H
def H():
    h,w = LI()
    s = SR(h)
    dp = [[0 for i in range(w)] for j in range(h)]
    dp[0][0] = 1
    for i in range(h-1):
        for j in range(w-1):
            if s[i+1][j] == ".":
                dp[i+1][j] += dp[i][j]
                dp[i+1][j] %= mod
            if s[i][j+1] == ".":
                dp[i][j+1] += dp[i][j]
                dp[i][j+1] %= mod
    for j in range(w-1):
        if s[h-1][j+1] == ".":
            dp[h-1][j+1] += dp[h-1][j]
            dp[h-1][j+1] %= mod
    for i in range(h-1):
        if s[i+1][w-1] == ".":
            dp[i+1][w-1] += dp[i][w-1]
            dp[i+1][w-1] %= mod
    print(dp[h-1][w-1])

#I
def J():
    n = I()
    p = list(map(float, input().split()))
    dp = [[0 for i in range(n+1)] for j in range(n+1)]
    dp[0][0] = 1
    for i in range(n):
        for j in range(n+1):
            dp[i+1][j] = p[i]*dp[i][j-1]+(1-p[i])*dp[i][j]

    ans = 0
    m = math.floor(n/2)
    for i in range(m+1,n+1):
        ans += dp[n][i]
    print(ans)

#J
def K():
    n = I()
    a = LI()
    d1 = a.count(1)
    d2 = a.count(2)
    d3 = a.count(3)
    dp = [[[0.0 for i in range(d3+1)] for j in range(d2+d3+1)] for k in range(n+1)]
    for k in range(d3+1):
        for j in range(d2+d3+1-k):
            for i in range(n+1-k-j):
                if i==j==k==0:continue
                l = i+j+k
                dp[i][j][k] = n
                if i:
                    dp[i][j][k] += i*dp[i-1][j][k]
                if j:
                    dp[i][j][k] += j*dp[i+1][j-1][k]
                if k:
                    dp[i][j][k] += k*dp[i][j+1][k-1]
                dp[i][j][k] /= l
    print(dp[d1][d2][d3])
#Solve
if __name__ == "__main__":
    K()

Submission status: Time Limit Exceeded

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = SR()
    return l
mod = 1000000007
sys.setrecursionlimit(1000000)

#A
def A():
    n = I()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n-2):
        dp[i+1] = min(dp[i+1],dp[i]+abs(h[i+1]-h[i]))
        dp[i+2] = min(dp[i+2],dp[i]+abs(h[i+2]-h[i]))
    dp[n-1] = min(dp[n-1],dp[n-2]+abs(h[n-1]-h[n-2]))
    print(dp[n-1])
#B
def B():
    n,k = LI()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n):
        for j in range(1,min(n-i,k+1)):
            dp[i+j] = min(dp[i+j],dp[i]+abs(h[i+j]-h[i]))
    print(dp[n-1])

#C
def C():
    n = I()
    v = LIR(n)
    dp = [[0,0,0] for i in range(n+1)]
    for i in range(n):
        dp[i+1][0] = max(dp[i+1][0],dp[i][1]+v[i][0],dp[i][2]+v[i][0])
        dp[i+1][1] = max(dp[i+1][1],dp[i][2]+v[i][1],dp[i][0]+v[i][1])
        dp[i+1][2] = max(dp[i+1][2],dp[i][0]+v[i][2],dp[i][1]+v[i][2])
    print(max(dp[n]))

#D
def D():
    n,W = LI()
    g = LIR(n)
    dp = [[0 for i in range(W+1)] for j in range(n+1)]
    for i in range(n):
        for j in range(W+1)[::-1]:
            w,v = g[i]
            if j >= w:
                dp[i+1][j] = max(dp[i][j],dp[i][j-w]+v)
            else:
                dp[i+1][j] = max(dp[i+1][j],dp[i][j])
    print(max(dp[n]))
#E
def E():
    n,W = LI()
    g = LIR(n)
    V = 100000
    dp = [[float("inf") for i in range(V+1)] for j in range(n+1)]
    dp[0][0] = 0
    for i in range(n):
        for j in range(V+1)[::-1]:
            w,v = g[i]
            if j >= v:
                dp[i+1][j] = min(dp[i][j],dp[i][j-v]+w)
            else:
                dp[i+1][j] = min(dp[i+1][j],dp[i][j])

    for i in range(V+1)[::-1]:
        if dp[n][i] <= W:
            print(i)
            quit()

#F
def F():
    s = S()
    t = S()
    dp = [[0 for i in range(len(t)+1)] for j in range(len(s)+1)]
    for i in range(len(s)):
        for j in range(len(t)):
            if s[i] == t[j]:
                dp[i+1][j+1] = max(dp[i][j]+1,dp[i+1][j],dp[i][j+1])
            else:
                dp[i+1][j+1] = max(dp[i+1][j],dp[i][j+1])

    i = len(s)
    j = len(t)
    ans = ""
    while i > 0 and j > 0:
        if dp[i][j] == dp[i-1][j]:i -= 1
        elif dp[i][j] == dp[i][j-1]:j -= 1
        else:
            i -= 1
            j -= 1
            ans = s[i]+ans
    print(ans)
#G
def G():
    def dfs(x):
        if dp[x] != None:
            return dp[x]
        if len(v[x]) == 0:
            dp[x] = 0
            return 0
        else:
            res = 0
            for y in v[x]:
                res = max(res,dfs(y)+1)
            dp[x] = res
            return dp[x]
    n,m = LI()
    dp = [None for i in range(n)]
    v = [[] for i in range(n)]
    for i in range(m):
        x,y = LI()
        v[x-1].append(y-1)
    for i in range(n):
        dfs(i)
    print(max(dp))

#H
def H():
    h,w = LI()
    s = SR(h)
    dp = [[0 for i in range(w)] for j in range(h)]
    dp[0][0] = 1
    for i in range(h-1):
        for j in range(w-1):
            if s[i+1][j] == ".":
                dp[i+1][j] += dp[i][j]
                dp[i+1][j] %= mod
            if s[i][j+1] == ".":
                dp[i][j+1] += dp[i][j]
                dp[i][j+1] %= mod
    for j in range(w-1):
        if s[h-1][j+1] == ".":
            dp[h-1][j+1] += dp[h-1][j]
            dp[h-1][j+1] %= mod
    for i in range(h-1):
        if s[i+1][w-1] == ".":
            dp[i+1][w-1] += dp[i][w-1]
            dp[i+1][w-1] %= mod
    print(dp[h-1][w-1])

#I
def J():
    n = I()
    p = list(map(float, input().split()))
    dp = [[0 for i in range(n+1)] for j in range(n+1)]
    dp[0][0] = 1
    for i in range(n):
        for j in range(n+1):
            dp[i+1][j] = p[i]*dp[i][j-1]+(1-p[i])*dp[i][j]

    ans = 0
    m = math.floor(n/2)
    for i in range(m+1,n+1):
        ans += dp[n][i]
    print(ans)

#J
def K():
    n = I()
    a = LI()
    d1 = a.count(1)
    d2 = a.count(2)
    d3 = a.count(3)
    dp = [[[0 for i in range(d3+1)] for j in range(d2+d3+1)] for k in range(n+1)]
    for k in range(d3+1):
        for j in range(d2+d3+1-k):
            for i in range(n+1-k-j):
                if i+j+k:
                    l = i+j+k
                    dp[i][j][k] = n
                    if i:
                        dp[i][j][k] += i*dp[i-1][j][k]
                    if j:
                        dp[i][j][k] += j*dp[i+1][j-1][k]
                    if k:
                        dp[i][j][k] += k*dp[i][j+1][k-1]
                    dp[i][j][k] /= l
    print(dp[d1][d2][d3])
#Solve
if __name__ == "__main__":
    K()

Submission status: Time Limit Exceeded

Solution code:
#!usr/bin/env python3
from collections import defaultdict
from collections import deque
from heapq import heappush, heappop
import sys
import math
import bisect
import random
def LI(): return list(map(int, sys.stdin.readline().split()))
def I(): return int(sys.stdin.readline())
def LS():return list(map(list, sys.stdin.readline().split()))
def S(): return list(sys.stdin.readline())[:-1]
def IR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = I()
    return l
def LIR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = LI()
    return l
def SR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = S()
    return l
def LSR(n):
    l = [None for i in range(n)]
    for i in range(n):l[i] = SR()
    return l
mod = 1000000007
sys.setrecursionlimit(1000000)

#A
def A():
    n = I()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n-2):
        dp[i+1] = min(dp[i+1],dp[i]+abs(h[i+1]-h[i]))
        dp[i+2] = min(dp[i+2],dp[i]+abs(h[i+2]-h[i]))
    dp[n-1] = min(dp[n-1],dp[n-2]+abs(h[n-1]-h[n-2]))
    print(dp[n-1])
#B
def B():
    n,k = LI()
    h = LI()
    dp = [float("inf") for i in range(n)]
    dp[0] = 0
    for i in range(n):
        for j in range(1,min(n-i,k+1)):
            dp[i+j] = min(dp[i+j],dp[i]+abs(h[i+j]-h[i]))
    print(dp[n-1])

#C
def C():
    n = I()
    v = LIR(n)
    dp = [[0,0,0] for i in range(n+1)]
    for i in range(n):
        dp[i+1][0] = max(dp[i+1][0],dp[i][1]+v[i][0],dp[i][2]+v[i][0])
        dp[i+1][1] = max(dp[i+1][1],dp[i][2]+v[i][1],dp[i][0]+v[i][1])
        dp[i+1][2] = max(dp[i+1][2],dp[i][0]+v[i][2],dp[i][1]+v[i][2])
    print(max(dp[n]))

#D
def D():
    n,W = LI()
    g = LIR(n)
    dp = [[0 for i in range(W+1)] for j in range(n+1)]
    for i in range(n):
        for j in range(W+1)[::-1]:
            w,v = g[i]
            if j >= w:
                dp[i+1][j] = max(dp[i][j],dp[i][j-w]+v)
            else:
                dp[i+1][j] = max(dp[i+1][j],dp[i][j])
    print(max(dp[n]))
#E
def E():
    n,W = LI()
    g = LIR(n)
    V = 100000
    dp = [[float("inf") for i in range(V+1)] for j in range(n+1)]
    dp[0][0] = 0
    for i in range(n):
        for j in range(V+1)[::-1]:
            w,v = g[i]
            if j >= v:
                dp[i+1][j] = min(dp[i][j],dp[i][j-v]+w)
            else:
                dp[i+1][j] = min(dp[i+1][j],dp[i][j])

    for i in range(V+1)[::-1]:
        if dp[n][i] <= W:
            print(i)
            quit()

#F
def F():
    s = S()
    t = S()
    dp = [[0 for i in range(len(t)+1)] for j in range(len(s)+1)]
    for i in range(len(s)):
        for j in range(len(t)):
            if s[i] == t[j]:
                dp[i+1][j+1] = max(dp[i][j]+1,dp[i+1][j],dp[i][j+1])
            else:
                dp[i+1][j+1] = max(dp[i+1][j],dp[i][j+1])

    i = len(s)
    j = len(t)
    ans = ""
    while i > 0 and j > 0:
        if dp[i][j] == dp[i-1][j]:i -= 1
        elif dp[i][j] == dp[i][j-1]:j -= 1
        else:
            i -= 1
            j -= 1
            ans = s[i]+ans
    print(ans)
#G
def G():
    def dfs(x):
        if dp[x] != None:
            return dp[x]
        if len(v[x]) == 0:
            dp[x] = 0
            return 0
        else:
            res = 0
            for y in v[x]:
                res = max(res,dfs(y)+1)
            dp[x] = res
            return dp[x]
    n,m = LI()
    dp = [None for i in range(n)]
    v = [[] for i in range(n)]
    for i in range(m):
        x,y = LI()
        v[x-1].append(y-1)
    for i in range(n):
        dfs(i)
    print(max(dp))

#H
def H():
    h,w = LI()
    s = SR(h)
    dp = [[0 for i in range(w)] for j in range(h)]
    dp[0][0] = 1
    for i in range(h-1):
        for j in range(w-1):
            if s[i+1][j] == ".":
                dp[i+1][j] += dp[i][j]
                dp[i+1][j] %= mod
            if s[i][j+1] == ".":
                dp[i][j+1] += dp[i][j]
                dp[i][j+1] %= mod
    for j in range(w-1):
        if s[h-1][j+1] == ".":
            dp[h-1][j+1] += dp[h-1][j]
            dp[h-1][j+1] %= mod
    for i in range(h-1):
        if s[i+1][w-1] == ".":
            dp[i+1][w-1] += dp[i][w-1]
            dp[i+1][w-1] %= mod
    print(dp[h-1][w-1])

#I
def J():
    n = I()
    p = list(map(float, input().split()))
    dp = [[0 for i in range(n+1)] for j in range(n+1)]
    dp[0][0] = 1
    for i in range(n):
        for j in range(n+1):
            dp[i+1][j] = p[i]*dp[i][j-1]+(1-p[i])*dp[i][j]

    ans = 0
    m = math.floor(n/2)
    for i in range(m+1,n+1):
        ans += dp[n][i]
    print(ans)

#J
def K():
    n = I()
    a = LI()
    d1 = a.count(1)
    d2 = a.count(2)
    d3 = a.count(3)
    dp = [[[0 for i in range(d3+1)] for j in range(d2+d3+1)] for k in range(n+1)]
    for k in range(d3+1):
        for j in range(d2+d3+1-k):
            for i in range(n+1-k-j):
                if i+j+k:
                    l = i+j+k
                    dp[i][j][k] = n/l
                    if i:
                        dp[i][j][k] += i/l*dp[i-1][j][k]
                    if j:
                        dp[i][j][k] += j/l*dp[i+1][j-1][k]
                    if k:
                        dp[i][j][k] += k/l*dp[i][j+1][k-1]

    print(dp[d1][d2][d3])
#Solve
if __name__ == "__main__":
    K()