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)

/**
 * author: derrick20
 * created: 8/27/20 2:18 PM
 */
import java.io.*;
import java.util.*;

public class Main {
    public static void main(String[] args) throws Exception {
        FastScanner sc = new FastScanner();
        PrintWriter out = new PrintWriter(System.out);

        int MAX = (int) 2e5; // todo change this!
        fact = new long[MAX + 1];
        invFact = new long[MAX + 1];
        fact[0] = 1;
        invFact[0] = 1;
        for (int i = 1; i <= MAX; i++) {
            fact[i] = ((long) i * fact[i - 1]) % mod;
            invFact[i] = inverse(fact[i]);
        }

        H = sc.nextInt();
        W = sc.nextInt();
        int N = sc.nextInt();
        nodes = new Node[N + 2];
        for (int i = 1; i <= N; i++) {
            nodes[i] = new Node(sc.nextInt(), sc.nextInt());
        }
        Node start = new Node(1, 1);
        Node end = new Node(H, W);
        nodes[0] = start;
        nodes[N + 1] = end;
        Arrays.sort(nodes, (a, b) -> {
            if (a.r <= b.r && a.c <= b.c) {
                return -1;
            } else {
                return a.r - b.r; // arbitrary
            }
        });
        long[] dp = new long[N + 2];
        dp[0] = 1;
        for (int v = 0; v <= N + 1; v++) {
            Node adj = nodes[v];
            for (int u = 0; u < v; u++) {
                Node node = nodes[u];
                if (node.r <= adj.r && node.c <= adj.c) {
                    int dr = adj.r - node.r;
                    int dc = adj.c - node.c;
                    ASSERT(dr >= 0 && dc >= 0);
                    long amt = (choose(dr + dc, dr) * dp[u]) % mod;
//                System.out.println("Branch from " +  node + " to " + adj + " = " + amt);
                    // apply a negative sign now!
                    dp[v] = (dp[v] + mod - amt) % mod;
                }
            }
        }

        long ans = (mod - dp[N + 1]) % mod;
        out.println(ans);
        out.close();
    }

    // todo SET THE MOD
    static long mod = (long) 1e9 + 7;
    static long[] fact, invFact;
    static Node[] nodes;
    static int H, W;

    static long choose(int n, int k) {
        ASSERT(k <= n, "Invalid choose");
        long ans = (fact[n] * invFact[n - k]) % mod;
        return (ans * invFact[k]) % mod;
    }

    static long inverse(long x) {
        return fastExpo(x, mod - 2);
    }

    static long fastExpo(long x, long k) {
        long ans = 1;
        while (k > 0) {
            if ((k & 1) > 0) {
                ans = (x * ans) % mod;
            }
            x = (x * x) % mod;
            k >>= 1;
        }
        return ans;
    }
    static class Node {
        int r, c;
        public Node(int rr, int cc) {
            r = rr; c = cc;
        }
        public String toString() {
            return "(" + r + ", " + c + ")";
        }
    }

    static class FastScanner {
        private int BS = 1 << 16;
        private char NC = (char) 0;
        private byte[] buf = new byte[BS];
        private int bId = 0, size = 0;
        private char c = NC;
        private double cnt = 1;
        private BufferedInputStream in;

        public FastScanner() {
            in = new BufferedInputStream(System.in, BS);
        }

        public FastScanner(String s) {
            try {
                in = new BufferedInputStream(new FileInputStream(new File(s)), BS);
            } catch (Exception e) {
                in = new BufferedInputStream(System.in, BS);
            }
        }

        private char getChar() {
            while (bId == size) {
                try {
                    size = in.read(buf);
                } catch (Exception e) {
                    return NC;
                }
                if (size == -1) return NC;
                bId = 0;
            }
            return (char) buf[bId++];
        }

        public int nextInt() {
            return (int) nextLong();
        }

        public int[] nextInts(int N) {
            int[] res = new int[N];
            for (int i = 0; i < N; i++) {
                res[i] = (int) nextLong();
            }
            return res;
        }

        public long[] nextLongs(int N) {
            long[] res = new long[N];
            for (int i = 0; i < N; i++) {
                res[i] = nextLong();
            }
            return res;
        }

        public long nextLong() {
            cnt = 1;
            boolean neg = false;
            if (c == NC) c = getChar();
            for (; (c < '0' || c > '9'); c = getChar()) {
                if (c == '-') neg = true;
            }
            long res = 0;
            for (; c >= '0' && c <= '9'; c = getChar()) {
                res = (res << 3) + (res << 1) + c - '0';
                cnt *= 10;
            }
            return neg ? -res : res;
        }

        public double nextDouble() {
            double cur = nextLong();
            return c != '.' ? cur : cur + nextLong() / cnt;
        }

        public double[] nextDoubles(int N) {
            double[] res = new double[N];
            for (int i = 0; i < N; i++) {
                res[i] = nextDouble();
            }
            return res;
        }

        public String next() {
            StringBuilder res = new StringBuilder();
            while (c <= 32) c = getChar();
            while (c > 32) {
                res.append(c);
                c = getChar();
            }
            return res.toString();
        }

        public String nextLine() {
            StringBuilder res = new StringBuilder();
            while (c <= 32) c = getChar();
            while (c != '\n') {
                res.append(c);
                c = getChar();
            }
            return res.toString();
        }

        public boolean hasNext() {
            if (c > 32) return true;
            while (true) {
                c = getChar();
                if (c == NC) return false;
                else if (c > 32) return true;
            }
        }
    }
    static void ASSERT(boolean assertion, String message) {
        if (!assertion) throw new AssertionError(message);
    }
    static void ASSERT(boolean assertion) {
        if (!assertion) throw new AssertionError();
    }
}