Problem:
Score : 100 points

Problem Statement

There is a grid with H horizontal rows and W vertical columns.
Let (i, j) denote the square at the i-th row from the top and the j-th column from the left.

In the grid, N Squares (r_1, c_1), (r_2, c_2), \ldots, (r_N, c_N) are wall squares, and the others are all empty squares.
It is guaranteed that Squares (1, 1) and (H, W) are empty squares.

Taro will start from Square (1, 1) and reach (H, W) by repeatedly moving right or down to an adjacent empty square.

Find the number of Taro's paths from Square (1, 1) to (H, W), modulo 10^9 + 7.

Constraints

All values in input are integers.

2 \leq H, W \leq 10^5

1 \leq N \leq 3000

1 \leq r_i \leq H

1 \leq c_i \leq W

Squares (r_i, c_i) are all distinct.

Squares (1, 1) and (H, W) are empty squares.

Input

Input is given from Standard Input in the following format:

H W N
r_1 c_1
r_2 c_2
:
r_N c_N

Output

Print the number of Taro's paths from Square (1, 1) to (H, W), modulo 10^9 + 7.

Sample Input 1

3 4 2
2 2
1 4

Sample Output 1

3

There are three paths as follows:

Sample Input 2

5 2 2
2 1
4 2

Sample Output 2

0

There may be no paths.

Sample Input 3

5 5 4
3 1
3 5
1 3
5 3

Sample Output 3

24

Sample Input 4

100000 100000 1
50000 50000

Sample Output 4

123445622

Be sure to print the count modulo 10^9 + 7.

Solution in Java (OpenJDK 11.0.6) (judge verdict: Wrong Answer)

import java.util.*;
import java.io.*;

public class Main {

    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 float nextFloat() {
            return Float.parseFloat(next());
        }

        public double nextDouble() {
            return Float.parseFloat(next());
        }

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

    static InputReader sc;
    static PrintWriter pw;

    static int mod = (int) (1e9 + 7);
    static int nax = (int) (2e5 + 5);

    static int[] fac;
    static int[] inv_fac;

    public static void main(String[] args) throws Exception {
        sc = new InputReader(System.in);
        pw = new PrintWriter(System.out);

        int h = sc.nextInt();
        int w = sc.nextInt();
        int n = sc.nextInt();

        fac = new int[nax];
        inv_fac = new int[nax];

        fac[0] = 1;
        inv_fac[0] = 1;

        for (int i = 1; i < nax; i++) {
            fac[i] = CPMath.multiply(fac[i-1], i);
            inv_fac[i] = CPMath.inverse(fac[i]);
        }

        Point[] points = new Point[n + 1];

        for (int i = 0; i < n; i++) {
            points[i] = new Point(sc.nextInt() - 1, sc.nextInt() - 1);
        }

        points[n] = new Point(h - 1, w - 1);

        Arrays.sort(points);

        int[] dp = new int[n + 1];

        for (int i = 0; i <= n; i++) {
            Point p = points[i];
            int numWays = CPMath.paths(p.x, p.y);

            for (int j = 0; j < i; j++) {
                Point p_j = points[j];
                if (p_j.x <= p.x && p_j.y <= p.y) {
                    numWays = CPMath.sub(numWays, CPMath.multiply(dp[j], CPMath.paths(p.x - p_j.x, p.y - p_j.y)));
                }
            }

            dp[i] = numWays;
        }

        pw.println(dp[n]);
        pw.close();
    }

    static class CPMath {
        static int add(int a, int b) {
            a += b;

            if (a >= mod) a -= mod;

            return a;
        }
        static int sub(int a, int b) {
            a -= b;
            if (a < 0) a+= mod;
            return a;
        }
        static int multiply(int a, int b) {
            return (int) ((long) (b) * a) % mod;
        }
        static int inverse(int a) {
            return power(a, mod - 2);
        }
        static int power(int a, int b) {
            int r = 1;

            while (b > 0) {
                if (b % 2 == 1) {
                    r = multiply(r, a);
                }

                a = multiply(a, a);
                b /= 2;
            }

            return r;
        }
        static int paths(int a, int b) {
            return binomial(a + b, b);
        }
        static int binomial(int a, int b) {
            if (a < 0 || b < 0 || b > a) return 0;
            return multiply(multiply(fac[a], inverse(fac[b])), inverse(fac[a - b]));
        }
    }

    static class Point implements Comparable<Point>{
        int x, y;

        Point(int x, int y) {
            this.x = x;
            this.y = y;
        }

        @Override
        public int compareTo(Point o) {
            return (this.x + this.y) - (o.x + o.y);
        }
    }
}