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 Rust (1.15.1) (judge verdict: Accepted)

// ---------- begin ModInt ----------
//https://github.com/kenkoooo/competitive-programming-rs/blob/master/src/math/mod_int.rs
//を参考にしています
#[allow(dead_code)]
mod modint {
    pub const MOD: u32 = 1_000_000_007;
    use std::ops::{Add, AddAssign, Div, DivAssign, Mul, MulAssign, Neg, Sub, SubAssign};
    #[derive(Clone, Copy)]
    pub struct ModInt<T: Copy + Clone>(pub T);
    type Num = u32;
    impl Add<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn add(self, other: ModInt<Num>) -> ModInt<Num> {
            let mut d = self.0 + other.0;
            if d >= MOD {
                d -= MOD;
            }
            ModInt(d)
        }
    }
    impl AddAssign<ModInt<Num>> for ModInt<Num> {
        fn add_assign(&mut self, other: ModInt<Num>) {
            *self = *self + other;
        }
    }
    impl Sub<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn sub(self, other: ModInt<Num>) -> ModInt<Num> {
            let mut d = self.0 + MOD - other.0;
            if d >= MOD {
                d -= MOD;
            }
            ModInt(d)
        }
    }
    impl SubAssign<ModInt<Num>> for ModInt<Num> {
        fn sub_assign(&mut self, other: ModInt<Num>) {
            *self = *self - other;
        }
    }
    impl Mul<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn mul(self, other: ModInt<Num>) -> ModInt<Num> {
            ModInt(((self.0 as u64) * (other.0 as u64) % (MOD as u64)) as u32)
        }
    }
    impl MulAssign<ModInt<Num>> for ModInt<Num> {
        fn mul_assign(&mut self, other: ModInt<Num>) {
            *self = *self * other;
        }
    }
    impl Div<ModInt<Num>> for ModInt<Num> {
        type Output = ModInt<Num>;
        fn div(self, other: ModInt<Num>) -> ModInt<Num> {
            self * other.pow(MOD - 2)
        }
    }
    impl DivAssign<ModInt<Num>> for ModInt<Num> {
        fn div_assign(&mut self, other: ModInt<Num>) {
            *self = *self / other;
        }
    }
    impl Neg for ModInt<Num> {
        type Output = ModInt<Num>;
        fn neg(self) -> ModInt<Num> {
            ModInt(if self.0 == 0 { 0 } else { MOD - self.0 })
        }
    }
    impl ModInt<Num> {
        pub fn new(v: u32) -> ModInt<Num> {
            ModInt(v % MOD)
        }
        pub fn pow(self, mut n: u32) -> ModInt<Num> {
            let mut t = ModInt::new(1);
            let mut s = self;
            while n > 0 {
                if (n & 1) == 1 {
                    t *= s;
                }
                s *= s;
                n >>= 1;
            }
            t
        }
    }
    pub struct Precalc {
        n: usize,
        inv: Vec<ModInt<Num>>,
        fact: Vec<ModInt<Num>>,
        ifact: Vec<ModInt<Num>>,
    }
    impl Precalc {
        pub fn new(n: usize) -> Precalc {
            let mut inv = vec![ModInt(1); n + 1];
            let mut fact = vec![ModInt(1); n + 1];
            let mut ifact = vec![ModInt(1); n + 1];
            for i in 1..(n + 1) {
                if i >= 2 {
                    inv[i] = -inv[(MOD as usize) % i] * ModInt(MOD / (i as u32));
                }
                fact[i] = ModInt(i as u32) * fact[i - 1];
                ifact[i] = inv[i] * ifact[i - 1];
            }
            Precalc {
                n: n,
                inv: inv,
                fact: fact,
                ifact: ifact,
            }
        }
        pub fn fact(&self, n: usize) -> ModInt<Num> {
            self.fact[n]
        }
        pub fn inv(&self, x: usize) -> ModInt<Num> {
            self.inv[x]
        }
        pub fn ifact(&self, x: usize) -> ModInt<Num> {
            self.ifact[x]
        }
        pub fn comb(&self, n: usize, k: usize) -> ModInt<Num> {
            if !(k <= n) {
                return ModInt(0);
            }
            self.fact[n] * self.ifact[k] * self.ifact[n - k]
        }
    }
    use std;
    impl std::fmt::Display for ModInt<Num> {
        fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
            write!(f, "{}", self.0)
        }
    }
}
// ---------- end ModInt ----------

use std::io::Read;
use modint::*;

fn run() {
    let mut s = String::new();
    std::io::stdin().read_to_string(&mut s).unwrap();
    let mut it = s.trim().split_whitespace();
    let h: usize = it.next().unwrap().parse().unwrap();
    let w: usize = it.next().unwrap().parse().unwrap();
    let n: usize = it.next().unwrap().parse().unwrap();
    let mut p = Vec::with_capacity(n + 1);
    for _ in 0..n {
        let r: usize = it.next().unwrap().parse().unwrap();
        let c: usize = it.next().unwrap().parse().unwrap();
        p.push((r, c));
    }
    p.push((h, w));
    p.sort();
    let mut dp = vec![ModInt(0); n + 1];
    let pc = Precalc::new(h + w);
    for (i, &(r, c)) in p.iter().enumerate() {
        let mut way = pc.comb(r + c - 2, r - 1);
        for (j, &(x, y)) in p[0..i].iter().enumerate() {
            if x <= r && y <= c {
                way -= dp[j] * pc.comb(r - x + c - y, r - x);
            }
        }
        dp[i] = way;
    }
    println!("{}", dp[n]);
}

fn main() {
    run();
}