Problem:
Score : 500 points

Problem Statement

We have a grid of squares with N rows and M columns. Let (i, j) denote the square at the i-th row from the top and j-th column from the left. We will choose K of the squares and put a piece on each of them.

If we place the K pieces on squares (x_1, y_1), (x_2, y_2), ..., and (x_K, y_K), the cost of this arrangement is computed as:

\sum_{i=1}^{K-1} \sum_{j=i+1}^K (|x_i - x_j| + |y_i - y_j|)

Find the sum of the costs of all possible arrangements of the pieces. Since this value can be tremendous, print it modulo 10^9+7.

We consider two arrangements of the pieces different if and only if there is a square that contains a piece in one of the arrangements but not in the other.

Constraints

2 \leq N \times M \leq 2 \times 10^5

2 \leq K \leq N \times M

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N M K

Output

Print the sum of the costs of all possible arrangements of the pieces, modulo 10^9+7.

Sample Input 1

2 2 2

Sample Output 1

8

There are six possible arrangements of the pieces, as follows:

((1,1),(1,2)), with the cost |1-1|+|1-2| = 1

((1,1),(2,1)), with the cost |1-2|+|1-1| = 1

((1,1),(2,2)), with the cost |1-2|+|1-2| = 2

((1,2),(2,1)), with the cost |1-2|+|2-1| = 2

((1,2),(2,2)), with the cost |1-2|+|2-2| = 1

((2,1),(2,2)), with the cost |2-2|+|1-2| = 1

The sum of these costs is 8.

Sample Input 2

4 5 4

Sample Output 2

87210

Sample Input 3

100 100 5000

Sample Output 3

817260251

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

Solution in Rust (1.42.0) (judge verdict: Accepted)

#![allow(unused, non_snake_case, dead_code, non_upper_case_globals)]
use {
    proconio::{marker::*, *},
    std::{cmp::*, collections::*, mem::*},
};

#[fastout]
fn main() {
    input! {
    n:i64,m:i64,k:i64,
     }
    let mut md = ModComb::new(MX, MOD);
    let mut xsum = 0;
    let mut ysum = 0;
    for i in 0..m {
        xsum += i * (m - i) * n * n;
    }
    xsum %= MOD;
    xsum *= md.comb(n * m - 2, k - 2);
    for i in 0..n {
        ysum += i * (n - i) * m * m;
    }
    ysum %= MOD;
    ysum *= md.comb(n * m - 2, k - 2);
    println!("{}", (xsum + ysum) % MOD);
}
const MOD: i64 = 1000000007;
const MX: usize = 210101;
pub fn mod_pow(x: i64, n: i64, m: i64) -> i64 {
    let mut res = 1;
    let mut x = x % m;
    let mut n = n;
    while n > 0 {
        if n & 1 == 1 {
            res = (res * x) % m;
        }
        x = (x * x) % m;
        n >>= 1;
    }
    res
}
struct ModComb {
    fact: Vec<i64>,
    fact_inv: Vec<i64>,
    n: usize,
    p: i64,
}
impl ModComb {
    fn initialize(&mut self) {
        let n = self.n;
        self.fact[0] = 1;
        for i in 1..n {
            self.fact[i] = (self.fact[i - 1] * i as i64) % self.p;
        }
        self.fact_inv[n - 1] = mod_pow(self.fact[n - 1], self.p - 2, self.p);
        for i in (0..n - 1).rev() {
            self.fact_inv[i] = (self.fact_inv[i + 1] * (i + 1) as i64) % self.p;
        }
    }
    fn new(max_n: usize, p: i64) -> ModComb {
        let mut ft = ModComb {
            fact: vec![0; max_n + 1],
            fact_inv: vec![0; max_n + 1],
            n: max_n + 1,
            p: p,
        };
        ModComb::initialize(&mut ft);
        ft
    }
    fn fact(&self, n: usize) -> i64 {
        self.fact[n]
    }
    fn comb(&self, n: i64, k: i64) -> i64 {
        if n < k {
            return 0;
        }
        (self.perm(n, k) * self.fact_inv[k as usize]) % self.p
    }
    fn perm(&self, n: i64, k: i64) -> i64 {
        if n < k {
            return 0;
        }
        self.fact[n as usize] * self.fact_inv[(n - k) as usize] % self.p
    }
    fn nsk(&self, n: i64, k: i64) -> i64 {
        if n < k {
            return 0;
        }
        let mut res = 0;
        for i in 0..k + 1 {
            let v = self.comb(k, i) * mod_pow(i, n, self.p) % self.p;
            if (k - i) % 2 == 1 {
                res = (res + self.p - v) % self.p;
            } else {
                res = (res + v) % self.p;
            }
        }
        return res * self.fact_inv[k as usize] % self.p;
    }
}