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)

#[allow(unused_macros)]
macro_rules ! input { ( source = $ s : expr , $ ( $ r : tt ) * ) => { let mut iter = $ s . split_whitespace ( ) ; let mut next = || { iter . next ( ) . unwrap ( ) } ; input_inner ! { next , $ ( $ r ) * } } ; ( $ ( $ r : tt ) * ) => { let stdin = std :: io :: stdin ( ) ; let mut bytes = std :: io :: Read :: bytes ( stdin . lock ( ) ) ; let mut next = move || -> String { bytes . by_ref ( ) . map ( | r | r . unwrap ( ) as char ) . skip_while ( | c | c . is_whitespace ( ) ) . take_while ( | c |! c . is_whitespace ( ) ) . collect ( ) } ; input_inner ! { next , $ ( $ r ) * } } ; }
#[allow(unused_macros)]
macro_rules ! input_inner { ( $ next : expr ) => { } ; ( $ next : expr , ) => { } ; ( $ next : expr , mut $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let mut $ var = read_value ! ( $ next , $ t ) ; input_inner ! { $ next $ ( $ r ) * } } ; ( $ next : expr , $ var : ident : $ t : tt $ ( $ r : tt ) * ) => { let $ var = read_value ! ( $ next , $ t ) ; input_inner ! { $ next $ ( $ r ) * } } ; }
#[allow(unused_macros)]
macro_rules ! read_value { ( $ next : expr , ( $ ( $ t : tt ) ,* ) ) => { ( $ ( read_value ! ( $ next , $ t ) ) ,* ) } ; ( $ next : expr , [ $ t : tt ; $ len : expr ] ) => { ( 0 ..$ len ) . map ( | _ | read_value ! ( $ next , $ t ) ) . collect ::< Vec < _ >> ( ) } ; ( $ next : expr , chars ) => { read_value ! ( $ next , String ) . chars ( ) . collect ::< Vec < char >> ( ) } ; ( $ next : expr , usize1 ) => { read_value ! ( $ next , usize ) - 1 } ; ( $ next : expr , $ t : ty ) => { $ next ( ) . parse ::<$ t > ( ) . expect ( "Parse error" ) } ; }

#[allow(unused_macros)]
macro_rules ! vec_md { ( $ init : expr ; $ ( $ dim : expr ) ;* ) => { vec_md ! ( $ ( $ dim ) ;* => $ init ) } ; ( $ head : expr ; $ ( $ tail : expr ) ;* => $ init : expr ) => { vec ! [ vec_md ! ( $ ( $ tail ) ;* => $ init ) ; $ head ] } ; ( $ head : expr => $ init : expr ) => { vec ! [ $ init ; $ head ] } ; }

fn main() {
    input! {
        h: usize, w: usize, mut n: usize,
        mut rc: [(usize, usize);n]
    }
    n += 2;
    rc.push((1,1));
    rc.push((h,w));
    rc.sort();

    let mut dp = vec_md![Co(0);n;2];
    let comb = Comb::new((h-1)+(w-1)+1);
    dp[0][0] = Co(1);
    for f in 0..n {
        for t in f+1..n {
            let (fy, fx) = rc[f];
            let (ty, tx) = rc[t];
            if fy <= ty && fx <= tx {
                let ptn = comb.of(ty-fy+tx-fx, tx-fx).into();
                dp[t][0] += dp[f][1] * ptn;
                dp[t][1] += dp[f][0] * ptn;
            }
        }
    }
    println!("{}", dp[n-1][1] - dp[n-1][0]);
}

#[allow(dead_code)]
const M: i64 = 1_000_000_007;

#[allow(dead_code)]
struct Comb {
    fac: Vec<i64>,
    finv: Vec<i64>,
    inv: Vec<i64>,
}
#[allow(dead_code)]
impl Comb {
    fn new(n: usize) -> Self {
        let mut fac = vec![0; n];
        let mut finv = vec![0; n];
        let mut inv = vec![0; n];
        fac[0] = 1;
        fac[1] = 1;
        finv[0] = 1;
        finv[1] = 1;
        inv[1] = 1;
        for i in 2..n {
            let i_i64 = i as i64;
            fac[i] = fac[i - 1] * i_i64 % M;
            inv[i] = M - inv[(M % i_i64) as usize] * (M / i_i64) % M;
            finv[i] = finv[i - 1] * inv[i] % M;
        }
        Comb {
            fac: fac,
            finv: finv,
            inv: inv,
        }
    }
    fn of(&self, n: usize, k: usize) -> i64 {
        if n < k {
            0
        } else {
            self.fac[n] * (self.finv[k] * self.finv[n - k] % M) % M
        }
    }
    #[doc = " combination with repetition"]
    fn rep(&self, n: usize, k: usize) -> i64 {
        self.of(n + k - 1, k)
    }
}

#[allow(dead_code)]
use std::ops::*;
#[allow(dead_code)]
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub struct Co(pub i64);
#[allow(dead_code)]
impl Co {
    pub fn pow(self, mut n: u64) -> Self {
        let mut a = self;
        let mut res = Co(1);
        while n > 0 {
            if n & 1 == 1 {
                res *= a;
            }
            a = a * a;
            n >>= 1;
        }
        res
    }
}
#[allow(dead_code)]
impl std::fmt::Display for Co {
    fn fmt(&self, f: &mut std::fmt::Formatter) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}
macro_rules ! impl_from { ( $ ( $ tpe : ident ) ,* ) => { $ ( impl From <$ tpe > for Co { fn from ( value : $ tpe ) -> Self { Co ( ( value as i64 % M + M ) % M ) } } ) * } }
impl_from!(i64, u64, usize);
macro_rules ! impl_binop { ( $ op : ident ,$ atr : ident ,$ assign : ident $ ( ,$ optr : ident ) * ) => { impl $ atr for Co { fn $ assign ( & mut self , other : Self ) { * self = self .$ op ( other ) ; } } $ ( impl $ optr for Co { type Output = Self ; fn $ op ( self , other : Self ) -> Self { Co :: from ( ( self . 0 ) .$ op ( other . 0 ) ) } } ) * } }
impl_binop!(add, AddAssign, add_assign, Add);
impl_binop!(sub, SubAssign, sub_assign, Sub);
impl_binop!(mul, MulAssign, mul_assign, Mul);
impl_binop!(div, DivAssign, div_assign);
#[allow(dead_code)]
impl Div for Co {
    type Output = Self;
    fn div(self, rhs: Self) -> Self {
        self * rhs.pow((M - 2) as u64)
    }
}