Problem:
Score : 800 points

Problem Statement

There are N children, numbered 1,2,\ldots,N.
In the next K days, we will give them some cookies.
In the i-th day, we will choose a_i children among the N with equal probability, and give one cookie to each child chosen. (We make these K choices independently.)

Let us define the happiness of the children as c_1 \times c_2 \times \ldots \times c_N, where c_i is the number of cookies received by Child i in the K days.
Find the expected happiness multiplied by \binom{N}{a_1} \times \binom{N}{a_2} \times \ldots \times \binom{N}{a_K} (we can show that this value is an integer), modulo (10^{9}+7).

Notes

\binom{n}{k} denotes the number of possible choices of k objects out of given distinct n objects, disregarding order.

Constraints

1 \leq N \leq 1000

1 \leq K \leq 20

1 \leq a_i \leq N

Input

Input is given from Standard Input in the following format:

N K
a_1 a_2 \ldots a_K

Output

Print the answer.

Sample Input 1

3 2
3 2

Sample Output 1

12

On the first day, all the children 1, 2, and 3 receive one cookie.

On the second day, two out of the three children 1, 2, and 3 receive one cookie.

Their happiness is 4 in any case, so the expected happiness is 4. Print this value multiplied by \binom{3}{3} \times \binom{3}{2}, which is 12.

Sample Input 2

856 16
399 263 665 432 206 61 784 548 422 313 848 478 827 26 398 63

Sample Output 2

337587117

Find the expected value multiplied by \binom{N}{a_1} \times \binom{N}{a_2} \times \ldots \times \binom{N}{a_K}, modulo (10^9+7).

Solution in Java8 (OpenJDK 1.8.0) (judge verdict: Accepted)

import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.IOException;
import java.io.UncheckedIOException;
import java.io.Closeable;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 */
public class Main {
    public static void main(String[] args) throws Exception {
        Thread thread = new Thread(null, new TaskAdapter(), "", 1 << 27);
        thread.start();
        thread.join();
    }

    static class TaskAdapter implements Runnable {
        @Override
        public void run() {
            InputStream inputStream = System.in;
            OutputStream outputStream = System.out;
            FastInput in = new FastInput(inputStream);
            FastOutput out = new FastOutput(outputStream);
            CCookieDistribution solver = new CCookieDistribution();
            solver.solve(1, in, out);
            out.close();
        }
    }

    static class CCookieDistribution {
        public void solve(int testNumber, FastInput in, FastOutput out) {
            int n = in.readInt();
            int k = in.readInt();
            int[] a = new int[k];
            Modular mod = new Modular(1e9 + 7);
            Power power = new Power(mod);
            Composite comp = new Composite(n, mod);
            Factorial fact = new Factorial(n, mod);

            int[][] ps = new int[k][];
            for (int i = 0; i < k; i++) {
                a[i] = in.readInt();
                ps[i] = new int[a[i] + 1];
                for (int j = 0; j <= a[i]; j++) {
                    ps[i][j] = mod.mul(comp.composite(n - j, a[i] - j), fact.invFact(j));
                    ps[i][j] = mod.mul(ps[i][j], comp.invComposite(n, a[i]));
                }
            }

            int[] p = new int[]{1};
            for (int i = 0; i < k; i++) {
                p = FastFourierTransform.multiplyMod(p, ps[i], mod.getMod());
            }
            int ans = p.length >= n ? p[n] : 0;
            ans = mod.mul(ans, fact.fact(n));
            for (int i = 0; i < k; i++) {
                ans = mod.mul(ans, comp.composite(n, a[i]));
            }

            out.println(ans);
        }

    }

    static class CachedLog2 {
        private static final int BITS = 16;
        private static final int LIMIT = 1 << BITS;
        private static final byte[] CACHE = new byte[LIMIT];

        static {
            int b = 0;
            for (int i = 0; i < LIMIT; i++) {
                while ((1 << (b + 1)) <= i) {
                    b++;
                }
                CACHE[i] = (byte) b;
            }
        }

        public static int ceilLog(int x) {
            int ans = floorLog(x);
            if ((1 << ans) < x) {
                ans++;
            }
            return ans;
        }

        public static int floorLog(int x) {
            return x < LIMIT ? CACHE[x] : (BITS + CACHE[x >>> BITS]);
        }

    }

    static class InverseNumber {
        int[] inv;

        public InverseNumber(int[] inv, int limit, Modular modular) {
            this.inv = inv;
            inv[1] = 1;
            int p = modular.getMod();
            for (int i = 2; i <= limit; i++) {
                int k = p / i;
                int r = p % i;
                inv[i] = modular.mul(-k, inv[r]);
            }
        }

        public InverseNumber(int limit, Modular modular) {
            this(new int[limit + 1], limit, modular);
        }

    }

    static class Factorial {
        int[] fact;
        int[] inv;
        Modular modular;

        public Modular getModular() {
            return modular;
        }

        public Factorial(int[] fact, int[] inv, InverseNumber in, int limit, Modular modular) {
            this.modular = modular;
            this.fact = fact;
            this.inv = inv;
            fact[0] = inv[0] = 1;
            for (int i = 1; i <= limit; i++) {
                fact[i] = modular.mul(fact[i - 1], i);
                inv[i] = modular.mul(inv[i - 1], in.inv[i]);
            }
        }

        public Factorial(int limit, Modular modular) {
            this(new int[limit + 1], new int[limit + 1], new InverseNumber(limit, modular), limit, modular);
        }

        public int fact(int n) {
            return fact[n];
        }

        public int invFact(int n) {
            return inv[n];
        }

    }

    static class Composite {
        final Factorial factorial;
        final Modular modular;

        public Composite(Factorial factorial) {
            this.factorial = factorial;
            this.modular = factorial.getModular();
        }

        public Composite(int limit, Modular modular) {
            this(new Factorial(limit, modular));
        }

        public int composite(int m, int n) {
            if (n > m) {
                return 0;
            }
            return modular.mul(modular.mul(factorial.fact(m), factorial.invFact(n)), factorial.invFact(m - n));
        }

        public int invComposite(int m, int n) {
            return modular.mul(modular.mul(factorial.invFact(m), factorial.fact(n)), factorial.fact(m - n));
        }

    }

    static class SequenceUtils {
        public static void swap(double[] data, int i, int j) {
            double tmp = data[i];
            data[i] = data[j];
            data[j] = tmp;
        }

    }

    static class Power {
        final Modular modular;

        public Power(Modular modular) {
            this.modular = modular;
        }

    }

    static class FastFourierTransform {
        private static double[][] realLevels = new double[30][];
        private static double[][] imgLevels = new double[30][];

        private static void prepareLevel(int i) {
            if (realLevels[i] == null) {
                realLevels[i] = new double[1 << i];
                imgLevels[i] = new double[1 << i];
                for (int j = 0, s = 1 << i; j < s; j++) {
                    realLevels[i][j] = Math.cos(Math.PI / s * j);
                    imgLevels[i][j] = Math.sin(Math.PI / s * j);
                }
            }
        }

        public static void dft(double[][] p, int m) {
            int n = 1 << m;

            int shift = 32 - Integer.numberOfTrailingZeros(n);
            for (int i = 1; i < n; i++) {
                int j = Integer.reverse(i << shift);
                if (i < j) {
                    SequenceUtils.swap(p[0], i, j);
                    SequenceUtils.swap(p[1], i, j);
                }
            }

            double[][] t = new double[2][1];
            for (int d = 0; d < m; d++) {
                int s = 1 << d;
                int s2 = s << 1;
                prepareLevel(d);
                for (int i = 0; i < n; i += s2) {
                    for (int j = 0; j < s; j++) {
                        int a = i + j;
                        int b = a + s;
                        mul(realLevels[d][j], imgLevels[d][j], p[0][b], p[1][b], t, 0);
                        sub(p[0][a], p[1][a], t[0][0], t[1][0], p, b);
                        add(p[0][a], p[1][a], t[0][0], t[1][0], p, a);
                    }
                }
            }
        }

        public static void idft(double[][] p, int m) {
            dft(p, m);

            int n = 1 << m;
            div(p[0][0], p[1][0], n, p, 0);
            div(p[0][n / 2], p[1][n / 2], n, p, n / 2);
            for (int i = 1, until = n / 2; i < until; i++) {
                double a = p[0][n - i];
                double b = p[1][n - i];
                div(p[0][i], p[1][i], n, p, n - i);
                div(a, b, n, p, i);
            }
        }

        private static void add(double r1, double i1, double r2, double i2, double[][] r, int i) {
            r[0][i] = r1 + r2;
            r[1][i] = i1 + i2;
        }

        private static void sub(double r1, double i1, double r2, double i2, double[][] r, int i) {
            r[0][i] = r1 - r2;
            r[1][i] = i1 - i2;
        }

        private static void mul(double r1, double i1, double r2, double i2, double[][] r, int i) {
            r[0][i] = r1 * r2 - i1 * i2;
            r[1][i] = r1 * i2 + i1 * r2;
        }

        private static void div(double r1, double i1, double r2, double[][] r, int i) {
            r[0][i] = r1 / r2;
            r[1][i] = i1 / r2;
        }

        public static int[] multiplyMod(int[] a, int[] b, int m) {
            return multiplyMod(a, a.length, b, b.length, m);
        }

        public static int[] multiplyMod(int[] a, int aLen, int[] b, int bLen, int m) {
            int need = aLen + bLen - 1;
            int n = 1 << CachedLog2.ceilLog(need);

            double[] aReal = new double[n];
            double[] aImag = new double[n];
            for (int i = 0; i < aLen; i++) {
                int x = (a[i] % m + m) % m;
                aReal[i] = x & ((1 << 15) - 1);
                aImag[i] = x >> 15;
            }
            dft(new double[][]{aReal, aImag}, CachedLog2.floorLog(n));

            double[] bReal = new double[n];
            double[] bImag = new double[n];
            for (int i = 0; i < bLen; i++) {
                int x = (b[i] % m + m) % m;
                bReal[i] = x & ((1 << 15) - 1);
                bImag[i] = x >> 15;
            }
            dft(new double[][]{bReal, bImag}, CachedLog2.floorLog(n));

            double[] faReal = new double[n];
            double[] faImag = new double[n];
            double[] fbReal = new double[n];
            double[] fbImag = new double[n];

            for (int i = 0; i < n; i++) {
                int j = (n - i) & (n - 1);

                double a1r = (aReal[i] + aReal[j]) / 2;
                double a1i = (aImag[i] - aImag[j]) / 2;
                double a2r = (aImag[i] + aImag[j]) / 2;
                double a2i = (aReal[j] - aReal[i]) / 2;

                double b1r = (bReal[i] + bReal[j]) / 2;
                double b1i = (bImag[i] - bImag[j]) / 2;
                double b2r = (bImag[i] + bImag[j]) / 2;
                double b2i = (bReal[j] - bReal[i]) / 2;

                faReal[i] = a1r * b1r - a1i * b1i - a2r * b2i - a2i * b2r;
                faImag[i] = a1r * b1i + a1i * b1r + a2r * b2r - a2i * b2i;

                fbReal[i] = a1r * b2r - a1i * b2i + a2r * b1r - a2i * b1i;
                fbImag[i] = a1r * b2i + a1i * b2r + a2r * b1i + a2i * b1r;
            }

            idft(new double[][]{faReal, faImag}, CachedLog2.floorLog(n));
            idft(new double[][]{fbReal, fbImag}, CachedLog2.floorLog(n));
            int[] res = new int[need];
            for (int i = 0; i < need; i++) {
                long aa = (long) (faReal[i] + 0.5);
                long bb = (long) (fbReal[i] + 0.5);
                long cc = (long) (faImag[i] + 0.5);
                res[i] = (int) ((aa % m + (bb % m << 15) + (cc % m << 30)) % m);
            }
            return res;
        }

    }

    static class FastOutput implements AutoCloseable, Closeable {
        private StringBuilder cache = new StringBuilder(10 << 20);
        private final Writer os;

        public FastOutput(Writer os) {
            this.os = os;
        }

        public FastOutput(OutputStream os) {
            this(new OutputStreamWriter(os));
        }

        public FastOutput println(int c) {
            cache.append(c).append('\n');
            return this;
        }

        public FastOutput flush() {
            try {
                os.append(cache);
                os.flush();
                cache.setLength(0);
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
            return this;
        }

        public void close() {
            flush();
            try {
                os.close();
            } catch (IOException e) {
                throw new UncheckedIOException(e);
            }
        }

        public String toString() {
            return cache.toString();
        }

    }

    static class FastInput {
        private final InputStream is;
        private byte[] buf = new byte[1 << 13];
        private int bufLen;
        private int bufOffset;
        private int next;

        public FastInput(InputStream is) {
            this.is = is;
        }

        private int read() {
            while (bufLen == bufOffset) {
                bufOffset = 0;
                try {
                    bufLen = is.read(buf);
                } catch (IOException e) {
                    bufLen = -1;
                }
                if (bufLen == -1) {
                    return -1;
                }
            }
            return buf[bufOffset++];
        }

        public void skipBlank() {
            while (next >= 0 && next <= 32) {
                next = read();
            }
        }

        public int readInt() {
            int sign = 1;

            skipBlank();
            if (next == '+' || next == '-') {
                sign = next == '+' ? 1 : -1;
                next = read();
            }

            int val = 0;
            if (sign == 1) {
                while (next >= '0' && next <= '9') {
                    val = val * 10 + next - '0';
                    next = read();
                }
            } else {
                while (next >= '0' && next <= '9') {
                    val = val * 10 - next + '0';
                    next = read();
                }
            }

            return val;
        }

    }

    static class Modular {
        int m;

        public int getMod() {
            return m;
        }

        public Modular(int m) {
            this.m = m;
        }

        public Modular(long m) {
            this.m = (int) m;
            if (this.m != m) {
                throw new IllegalArgumentException();
            }
        }

        public Modular(double m) {
            this.m = (int) m;
            if (this.m != m) {
                throw new IllegalArgumentException();
            }
        }

        public int valueOf(long x) {
            x %= m;
            if (x < 0) {
                x += m;
            }
            return (int) x;
        }

        public int mul(int x, int y) {
            return valueOf((long) x * y);
        }

        public String toString() {
            return "mod " + m;
        }

    }
}