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.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.PrintWriter;
import java.util.InputMismatchException;
import java.io.IOException;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        FastScanner in = new FastScanner(inputStream);
        PrintWriter out = new PrintWriter(outputStream);
        TaskJ solver = new TaskJ();
        solver.solve(1, in, out);
        out.close();
    }

    static class TaskJ {
        public void solve(int testNumber, FastScanner in, PrintWriter out) {
            int n = in.nextInt();
            double[][][] dp = new double[n + 1][n + 1][n + 1];

            int c[] = new int[4];
            for (int i = 0; i < n; i++)
                c[in.nextInt()]++;

            for (int c3 = 0; c3 <= n; c3++) {
                for (int c2 = 0; (c2 + c3) <= n; c2++) {
                    for (int c1 = (c2 | c3) == 0 ? 1 : 0; (c1 + c2 + c3) <= n; c1++) {
                        int s = c1 + c2 + c3;
                        dp[c1][c2][c3] = (double) n / s;
                        if (c3 > 0)
                            dp[c1][c2][c3] += dp[c1][c2 + 1][c3 - 1] * c3 / s;
                        if (c2 > 0)
                            dp[c1][c2][c3] += dp[c1 + 1][c2 - 1][c3] * c2 / s;
                        if (c1 > 0)
                            dp[c1][c2][c3] += dp[c1 - 1][c2][c3] * c1 / s;
                    }
                }
            }

            out.println(dp[c[1]][c[2]][c[3]]);
        }

    }

    static class FastScanner {
        private InputStream in;
        private byte[] buffer = new byte[1024];
        private int bufPointer;
        private int bufLength;

        public FastScanner(InputStream in) {
            this.in = in;
        }

        private int readByte() {
            if (bufPointer >= bufLength) {
                if (bufLength == -1)
                    throw new InputMismatchException();

                bufPointer = 0;
                try {
                    bufLength = in.read(buffer);
                } catch (IOException e) {
                    throw new InputMismatchException();
                }

                if (bufLength <= 0)
                    return -1;
            }
            return buffer[bufPointer++];
        }

        private static boolean isSpaceChar(int c) {
            return c == ' ' || c == '\n' || c == '\r' || c == '\t' || c == -1;
        }

        public long nextLong() {
            long n = 0;

            int b = readByte();
            while (isSpaceChar(b))
                b = readByte();

            boolean minus = (b == '-');
            if (minus)
                b = readByte();

            while (b >= '0' && b <= '9') {
                n *= 10;
                n += b - '0';
                b = readByte();
            }

            if (!isSpaceChar(b))
                throw new NumberFormatException();

            return minus ? -n : n;
        }

        public int nextInt() {
            long n = nextLong();

            if (n < Integer.MIN_VALUE || n > Integer.MAX_VALUE)
                throw new NumberFormatException();

            return (int) n;
        }

    }
}