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.io.*;
import java.util.*;

public class Main {
	
	static int N;
	static int[] num;
	static double[][][] dp;
	static boolean[][][] solved;
		
	public static void main(String[] args) throws IOException {
		FastScanner sc = new FastScanner(System.in);
		
		N = sc.nextInt();
		num = new int[4];
		for(int i = 0; i < N; i++){
			num[sc.nextInt()]++;
		}
		
		dp = new double[N+1][N+1][N+1];
		solved = new boolean[N+1][N+1][N+1];
		
		System.out.println(solve(num[1], num[2], num[3]));
		
	}//end main
	
	public static double solve(int i, int j, int k){
		if(solved[i][j][k]) return dp[i][j][k];
		if(i == 0 && j == 0 && k == 0) return 0;
		
		double res = 0;
		
		if(k >= 1){
			res += (double)k * (1 + solve(i, j+1, k-1)) / N;
		}
		
		if(j >= 1){
			res += (double)j * (1 + solve(i+1, j-1, k)) / N;
		}
		
		if(i >= 1){
			res += (double)i * (1 + solve(i-1, j, k)) / N;
		}
		
		if(!(i == num[1] && j == num[2] && k == num[3])){
			res += (double)(N - (i + j + k)) / N;
			res *= (double)N / (i + j + k);
		}
		
		dp[i][j][k] = res;
		solved[i][j][k] = true;
		return dp[i][j][k];
	}
	
}//end Main

class FastScanner {
    private InputStream in;
    private final byte[] buffer = new byte[1024];
    private int ptr = 0;
    private int buflen = 0;
    public FastScanner(InputStream in) {
		this.in = in;
	}
    private boolean hasNextByte() {
        if (ptr < buflen) {
            return true;
        }else{
            ptr = 0;
            try {
                buflen = in.read(buffer);
            } catch (IOException e) {
                e.printStackTrace();
            }
            if (buflen <= 0) {
                return false;
            }
        }
        return true;
    }
    private int readByte() {
    	if (hasNextByte()) return buffer[ptr++];
    	else return -1;
    }
    private static boolean isPrintableChar(int c){
    	return 33 <= c && c <= 126;
    }
    public boolean hasNext() {
    	while(hasNextByte() && !isPrintableChar(buffer[ptr]))
    		ptr++; return hasNextByte();
    }
    public String next() {
        if (!hasNext()) throw new NoSuchElementException();
        StringBuilder sb = new StringBuilder();
        int b = readByte();
        while(isPrintableChar(b)) {
            sb.appendCodePoint(b);
            b = readByte();
        }
        return sb.toString();
    }
    public long nextLong() {
        if (!hasNext()) throw new NoSuchElementException();
        long n = 0;
        boolean minus = false;
        int b = readByte();
        if (b == '-') {
            minus = true;
            b = readByte();
        }
        if (b < '0' || '9' < b) {
            throw new NumberFormatException();
        }
        while(true){
            if ('0' <= b && b <= '9') {
                n *= 10;
                n += b - '0';
            }else if(b == -1 || !isPrintableChar(b)){
                return minus ? -n : n;
            }else{
                throw new NumberFormatException();
            }
            b = readByte();
        }
    }
    public int nextInt() {
        long nl = nextLong();
        if (nl < Integer.MIN_VALUE || nl > Integer.MAX_VALUE) throw new NumberFormatException();
        return (int) nl;
    }
    public double nextDouble() {
    	return Double.parseDouble(next());
    }
}