Problem: 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}.

Examples

Input

3
1 1 1


Output

5.5


Input

1
3


Output

3


Input

2
1 2


Output

4.5


Input

10
1 3 2 3 3 2 3 2 1 3


Output

54.48064457488221

Solution (in JAVA):

Solution:
import java.util.*;
import java.lang.*;
import java.io.*;
class Main{
    static FastScanner sc = new FastScanner(System.in);
    static PrintWriter out = new PrintWriter(System.out);
    static double dp[][][]=new double[301][301][301];
    public static double Ok(int x,int y,int z,int n){
        if(x==0 && y==0 && z==0) return 0;
        if(x<0 || y<0 || z<0) return 0;

        if(dp[x][y][z]!=-1) return dp[x][y][z];

        double op=n+x*Ok(x-1,y,z,n)+y*Ok(x+1,y-1,z,n)+z*Ok(x,y+1,z-1,n);

        return dp[x][y][z]=op/(x+y+z);
    }
    public static void Solve() throws Exception{

        int n =sc.nextInt();
        int x=0,y=0,z=0;
        for(int i=0;i<n;i++) {
            int x1 =sc.nextInt();
            if(x1==1) x++;
            if(x1==2) y++;
            if(x1==3) z++;
        }
        for(int i=0;i<301;i++){
            for(int j=0;j<301;j++){
                Arrays.fill(dp[i][j],-1);
            }
        }
        // out.println(x+" "+y+" "+z);
        out.println(Ok(x,y,z,n));


    }
    public static void main(String args[]) throws Exception{
        
        Solve();
        out.close();
    }  
    
}
class FastScanner
{

    final private int BUFFER_SIZE = 1 << 17;
    private DataInputStream din;
    private byte[] buffer;
    private int bufferPointer, bytesRead;

    public FastScanner(InputStream in)
    {
        din = new DataInputStream(in);
        buffer = new byte[BUFFER_SIZE];
        bufferPointer = bytesRead = 0;
    }
    public String nextLine() throws Exception
    {
        StringBuffer sb = new StringBuffer("");
        byte c = read();
        while (c <= ' ') c = read();
        do
        {
            sb.append((char)c);
            c = read();
        }
        while(c > ' ');
        return sb.toString();
    }

    public char nextChar() throws Exception
    {
        byte c = read();
        while(c <= ' ') c = read();
        return (char)c;
    }

    public int nextInt() throws Exception
    {
        int ret = 0;
        byte c = read();
        while (c <= ' ') c = read();
        boolean neg = c == '-';
        if (neg) c = read();
        do
        {
            ret = ret * 10 + c - '0';
            c = read();
        }
        while (c > ' ');
        if (neg) return -ret;
        return ret;
    }

    public long nextLong() throws Exception
    {
        long ret = 0;
        byte c = read();
        while (c <= ' ') c = read();
        boolean neg = c == '-';
        if (neg) c = read();
        do
        {
            ret = ret * 10 + c - '0';
            c = read();
        }
        while (c > ' ');
        if (neg) return -ret;
        return ret;
    } 

    private void fillBuffer() throws Exception
    {
        bytesRead = din.read(buffer, bufferPointer = 0, BUFFER_SIZE);
        if (bytesRead == -1) buffer[0] = -1;
    }

    private byte read() throws Exception
    {
        if (bufferPointer == bytesRead) fillBuffer();
        return buffer[bufferPointer++];
    }
    public double nextDouble() throws Exception
    {
        double ret = 0, div = 1;
        byte c = read();
        while (c <= ' ')
            c = read();
        boolean neg = (c == '-');
        if (neg)
            c = read();

        do
        {
            ret = ret * 10 + c - '0';
        }
        while ((c = read()) >= '0' && c <= '9');

        if (c == '.')
        {
            while ((c = read()) >= '0' && c <= '9')
            {
                ret += (c - '0') / (div *= 10);
            }
        }

        if (neg)
            return -ret;
        return ret;
    }

    
}