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.util.*;
import java.lang.*;
import java.io.*;
import java.math.*;
import java.util.concurrent.*;

public class Main {
    //---------------------------------------------------------------------------

    static int M = 51;
    static double[][][] dp;
    static long[][] C;
    static int n, m;
    static int[] a;

    static double f(int i, int s, int mx) {
        if (i >= m) {
            if (s != 0) {
                return 0;
            } else {
                return mx;
            }
        }
        double ret = dp[i][s][mx];
        if (Double.compare(ret, -1) != 0) {
            return ret;
        }
        ret = 0;
        double p = 1.0;
        for (int j = 0; j <= s; j++) {
            int l = (j + a[i] - 1) / a[i];
            ret += C[s][j] * p * f(i + 1, s - j, Math.max(mx, l));
            p /= m;
        }
        dp[i][s][mx] = ret;
        return ret;
    }

    public static void main(String[] args) throws Exception {
        InputReader in = new InputReader(System.in);
        PrintWriter out = new PrintWriter(System.out);
        //-----------------------------------------------------------------------
        C = new long[M][M];
        C[0][0] = 1;
        for (int i = 1; i < M; i++) {
            C[i][0] = 1;
            for (int j = 1; j <= i; j++) {
                C[i][j] = C[i - 1][j - 1] + C[i - 1][j];
            }
        }
        n = in.nextInt();
        m = in.nextInt();
        a = new int[m];
        for (int i = 0; i < m; i++) {
            a[i] = in.nextInt();
        }
        dp = new double[M][M][M];
        for (int i = 0; i < M; i++) {
            for (int j = 0; j < M; j++) {
                Arrays.fill(dp[i][j], -1);
            }
        }
        double res = f(0, n, 0);
        out.printf("%.15f\n", res);
        //-----------------------------------------------------------------------
        out.close();
    }

    //---------------------------------------------------------------------------
    static void shuffleArray(int[] ar) {
        Random rnd = ThreadLocalRandom.current();
        for (int i = ar.length - 1; i > 0; i--) {
            int index = rnd.nextInt(i + 1);
            int a = ar[index];
            ar[index] = ar[i];
            ar[i] = a;
        }
    }

    static class Pair<F, S> implements Comparable<Pair<F, S>> {

        F first;
        S second;

        public Pair() {

        }

        public Pair(F f, S s) {
            this.first = f;
            this.second = s;
        }

        public int compareTo(Pair<F, S> p) {
            int ret = ((Comparable<F>) first).compareTo(p.first);
            if (ret == 0) {
                ret = ((Comparable<S>) second).compareTo(p.second);
            }
            return ret;
        }
    }

    static class BitMask {
        int mask;

        public BitMask() {
            mask = 0;
        }
        public BitMask(BitMask b) {
            mask = b.getInt();
        }
        public BitMask(int i) {
            mask = i;
        }

        public int getInt() {
            return mask;
        }

        public int get(int i) {
            return (mask >> i) & 1;
        }

        public void set(int i, int v) {
            if (v == 0) {
                mask = mask & ~(1 << i);
            } else {
                mask = mask | (1 << i);
            }
        }

        public void set(int v) {
            for (int i = 0; i < 32; i++) {
                set(i, v);
            }
        }

        public void flip(int i) {
            mask = mask ^ (1 << i);
        }

        public void flip() {
            for (int i = 0; i < 31; i++) {
                flip(i);
            }
        }

        public int count() {
            int ret = 0;
            for (int i = 0; i < 32; i++) {
                if (get(i) == 1) {
                    ret++;
                }
            }
            return ret;
        }

        public static int grayCode(int i) {
            return i ^ (i >> 1);
        }
    }

    static class InputReader {

        public BufferedReader reader;
        public StringTokenizer tokenizer;

        public InputReader(InputStream stream) {
            reader = new BufferedReader(new InputStreamReader(stream), 32768);
            tokenizer = null;
        }

        public String next() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return tokenizer.nextToken();
        }

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

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

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

        public char[] nextCharArray() {
            return next().toCharArray();
        }

        public boolean hasNext() {
            while (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    String s = reader.readLine();
                    if (s == null) {
                        return false;
                    }
                    tokenizer = new StringTokenizer(s);
                } catch (IOException e) {
                    throw new RuntimeException(e);
                }
            }
            return true;
        }

        public void skipLine() {
            try {
                tokenizer = null;
                reader.readLine();
            } catch (IOException e) {
                throw new RuntimeException(e);
            }
        }
    }
}