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.BufferedWriter;
import java.io.FileReader;
import java.io.FileWriter;
import java.io.IOException;
import java.io.InputStreamReader;
import java.io.OutputStreamWriter;
import java.io.PrintWriter;
import java.util.Arrays;
import java.util.Locale;
import java.util.StringTokenizer;

@SuppressWarnings("unchecked")
public class Main implements Runnable {
	private static final String TASKNAME = "a";
	int n, m;
	int[] a;
	double[][] c;

	private void solve() throws Exception {
		n = nextInt();
		m = nextInt();
		a = new int[m];
		for (int i = 0; i < m; ++i) {
			a[i] = nextInt();
		}
		c = new double[Math.max(n, m) + 1][Math.max(n, m) + 1];
		for (int i = 0; i < c.length; ++i) {
			c[i][0] = 1;
			for (int j = 1; j <= i; ++j) {
				c[i][j] = c[i - 1][j - 1] + c[i - 1][j];
			}
		}
		double[] f = new double[c.length];
		f[0] = 1;
		for (int i = 1; i < f.length; ++i) {
			f[i] = f[i - 1] * i;
		}

		double[][][][] dp = new double[2][m + 1][n + 1][n + 1];
		clear(dp, 0);
		dp[0][0][0][0] = 1;
		for (int i = 0; i < m; ++i) {
			int x = (i + 1) % 2;
			clear(dp, x);
			for (int j = 0; j <= m; ++j) {
				for (int k = 0; k <= n; ++k) {
					for (int l = 0; l <= n; ++l) {
						if (dp[i & 1][j][k][l] == 0) {
							continue;
						}
						for (int am = 0; am + l <= n; ++am) {
							int max = Math.max(k, (am + a[i] - 1) / a[i]);
							dp[x][j + am > 0 ? 1 : 0][max][l + am] += dp[i & 1][j][k][l]
									* c[n - l][am];
						}
					}
				}
			}
		}

		double answer = 0;
		for (int i = 0; i <= m; ++i) {
			for (int j = 0; j <= n; ++j) {
				answer += dp[m & 1][i][j][n] * Math.pow(1.0 / m, n) 
				/ f[i] * j;
			}
		}

		println(answer);
	}

	private void clear(double[][][][] dp, int i) {
		for (int j = 0; j <= m; ++j) {
			for (int k = 0; k <= n; ++k) {
				for (int l = 0; l <= n; ++l) {
					dp[i][j][k][l] = 0;
				}
			}
		}
	}

	private BufferedReader reader;
	private PrintWriter writer;
	private StringTokenizer tokenizer;

	public void run() {
		try {
//			reader = new BufferedReader(new FileReader(TASKNAME + ".in"));
//			writer = new PrintWriter(new BufferedWriter(new FileWriter(TASKNAME
//					+ ".out")));
			reader = new BufferedReader(new InputStreamReader(System.in));
			writer = new PrintWriter(new BufferedWriter(new OutputStreamWriter(System.out)));

			solve();

			reader.close();
			writer.close();
		} catch (Throwable e) {
			throw new AssertionError(e);
		}
	}

	private void print(final Object o) {
		writer.print(o);
	}

	private void println(final Object o) {
		writer.println(o);
	}

	private void printf(final String format, final Object... o) {
		writer.printf(format, o);
	}

	private double nextDouble() throws IOException {
		return Double.parseDouble(nextToken());
	}

	private int nextInt() throws IOException {
		return Integer.parseInt(nextToken());
	}

	private long nextLong() throws IOException {
		return Long.parseLong(nextToken());
	}

	private String nextToken() throws IOException {
		while (tokenizer == null || !tokenizer.hasMoreTokens()) {
			tokenizer = new StringTokenizer(reader.readLine());
		}
		return tokenizer.nextToken();
	}

	public static void main(String[] args) throws InterruptedException {
		final long startTime = System.currentTimeMillis();
		Locale.setDefault(Locale.US);
		final Thread thread = new Thread(null, new Main(), "", 1 << 23);
		thread.start();
		thread.join();
		System.err.printf("%.3f\n",
				(System.currentTimeMillis() - startTime) * 0.001);
	}
}