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 (fact, invfact) = fact_inv_table(MX, MOD);
    let mut xsum = 0;
    let mut ysum = 0;
    for i in 0..m {
        xsum += i * (m - i) * n * n;
    }
    xsum %= MOD;
    xsum *= mod_comb(n * m - 2, k - 2, MOD, &fact);
    for i in 0..n {
        ysum += i * (n - i) * m * m;
    }
    ysum %= MOD;
    ysum *= mod_comb(n * m - 2, k - 2, MOD, &fact);
    println!("{}", (xsum + ysum) % MOD);
}
const MOD: i64 = 1000000007;
const MX: usize = 1010101;
/// (gcd, x, y)
pub fn extgcd(a: i64, b: i64) -> (i64, i64, i64) {
    if b == 0 {
        (a, 1, 0)
    } else {
        let (gcd, x, y) = extgcd(b, a % b);
        (gcd, y, x - (a / b) * y)
    }
}
/// x ^ n % m
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
}
pub fn mod_inverse(a: i64, m: i64) -> i64 {
    let (_, x, _) = extgcd(a as i64, m as i64);
    ((m as i64 + x) as i64 % m) % m
}
pub fn fact_table(len: usize, m: i64) -> Vec<i64> {
    let mut res = vec![1; len + 1];
    for i in 1..len + 1 {
        res[i] = (i as i64 * res[i - 1]) % m;
    }
    res
}
/// Factorial and Inverse factorial table
pub fn fact_inv_table(size: usize, m: i64) -> (Vec<i64>, Vec<i64>) {
    let mut fact = vec![1; size];
    let mut fact_inv = vec![1; size];
    for i in 2..size {
        fact[i] = fact[i - 1] * i as i64 % m;
        fact_inv[i] = m - ((m / i as i64) * fact_inv[(m % i as i64) as usize] % m);
    }
    for i in 1..size {
        fact_inv[i] = fact_inv[i - 1] * fact_inv[i] % m;
    }
    (fact, fact_inv)
}
/// (a mod p, e when n! = a p\^e)
pub fn mod_fact(n: i64, p: i64, fact: &[i64]) -> (i64, i64) {
    if n == 0 {
        (1, 0)
    } else {
        let (a, b) = mod_fact(n / p, p, fact);
        let pow = b + n / p;
        if n / p % 2 != 0 {
            (a * (p - fact[(n % p) as usize]) % p, pow)
        } else {
            (a * fact[(n % p) as usize] % p, pow)
        }
    }
}
/// C(n, k) % p
pub fn mod_comb(n: i64, k: i64, p: i64, fact: &[i64]) -> i64 {
    if n < k {
        0
    } else {
        let (a1, e1) = mod_fact(n, p, fact);
        let (a2, e2) = mod_fact(k, p, fact);
        let (a3, e3) = mod_fact(n - k, p, fact);
        if e1 > e2 + e3 {
            0
        } else {
            a1 * mod_inverse(a2 * a3 % p, p) % p
        }
    }
}
/// H(n, k) % p
pub fn mod_comb_repetition(n: i64, k: i64, p: i64, fact: &[i64]) -> i64 {
    mod_comb(n - 1 + k, n - 1, p, fact)
}