Problem: There are n students living in the campus. Every morning all students wake up at the same time and go to wash. There are m rooms with wash basins. The i-th of these rooms contains ai wash basins. Every student independently select one the rooms with equal probability and goes to it. After all students selected their rooms, students in each room divide into queues by the number of wash basins so that the size of the largest queue is the least possible. Calculate the expected value of the size of the largest queue among all rooms.

Input

The first line contains two positive integers n and m (1 ≤ n, m ≤ 50) — the amount of students and the amount of rooms. The second line contains m integers a1, a2, ... , am (1 ≤ ai ≤ 50). ai means the amount of wash basins in the i-th room.

Output

Output single number: the expected value of the size of the largest queue. Your answer must have an absolute or relative error less than 10 - 9.

Examples

Input

1 1
2


Output

1.00000000000000000000


Input

2 2
1 1


Output

1.50000000000000000000


Input

2 3
1 1 1


Output

1.33333333333333350000


Input

7 5
1 1 2 3 1


Output

2.50216960000000070000

Solution (in JAVA):

Solution:
import java.io.BufferedReader;
import java.io.FileNotFoundException;
import java.io.FileReader;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.StringTokenizer;

public class Solve8 {

    public static void main(String[] args) throws IOException {
        PrintWriter pw = new PrintWriter(System.out);
        new Solve8().solve(pw);
        pw.flush();
        pw.close();
    }

    public void solve(PrintWriter pw) throws IOException {
        FastReader sc = new FastReader();
        n = sc.nextInt();
        m = sc.nextInt();
        a = new int[m];
        for (int i = 0; i < m; i++) {
            a[i] = sc.nextInt();
        }
        _4dDP(m, n + 1, 2, n + 1);
        pascalTriangle();
        double s = 0;
        p = Math.pow(m, n);
        for (int i = 1; i <= n; i++) {
            s += i * dp(0, n, 0, i);
        }
        pw.println(s);
    }
    int n, m;
    int[] a;
    double p;
    double[][][][] dp;

    public double dp(int i, int s, int f, int q) {
        if (i == m) {
            return s == 0 && f > 0 ? 1 / p : 0;
        }
        if (dp[i][s][f][q] != -1) {
            return dp[i][s][f][q];
        }
        double y = 0;
        for (int j = 0; s - j >= 0 && j <= a[i] * q; j++) {
            int x = (j / a[i] == q ? 1 : 0) + (j / a[i] + 1 == q && j % a[i] > 0 ? 1 : 0);
            y += pascalTriangle[s][j] * dp(i + 1, s - j, f | x, q);
        }
        return dp[i][s][f][q] = y;
    }
    long[][] pascalTriangle;

    public void pascalTriangle() {
        final int MAX = (int) 50;
        pascalTriangle = new long[MAX + 1][];
        for (int i = 0; i <= MAX; i++) {
            pascalTriangle[i] = new long[i + 1];
            pascalTriangle[i][0] = pascalTriangle[i][i] = 1;
            for (int j = 1; j < i; j++) {
                pascalTriangle[i][j] = pascalTriangle[i - 1][j - 1] + pascalTriangle[i - 1][j];
            }
        }
    }

    public void _4dDP(int n, int m, int k, int v) {
        dp = new double[n][m][k][v];
        for (double[][][] _3dArr : dp) {
            for (double[][] _2dArr : _3dArr) {
                for (double[] row : _2dArr) {
                    Arrays.fill(row, -1);
                }
            }
        }
    }

    static class FastReader {

        BufferedReader br;
        StringTokenizer st;

        public FastReader() {
            br = new BufferedReader(new InputStreamReader(System.in));
        }

        public FastReader(String s) {
            try {
                br = new BufferedReader(new FileReader(s));
            } catch (FileNotFoundException e) {
            }
        }

        public String next() {
            if (st == null || !st.hasMoreTokens()) {
                try {
                    st = new StringTokenizer(br.readLine());
                } catch (Exception e) {
                }
            }
            return st.nextToken();
        }

        public int nextInt() {
            return Integer.parseInt(next());
        }

        public long nextLong() {
            return Long.parseLong(next());
        }

        public double nextDouble() {
            return Double.parseDouble(next());
        }

        public String nextLine() {
            try {
                return br.readLine();
            } catch (Exception e) {
            }
            return null;
        }

        public boolean hasNext() throws IOException {
            if (st != null && st.hasMoreTokens()) {
                return true;
            }
            String s = br.readLine();
            if (s == null || s.isEmpty()) {
                return false;
            }
            st = new StringTokenizer(s);
            return true;
        }
    }
}