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 (1.15.1) (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 ( std :: io :: BufReader :: new ( 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 , $ 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(dead_code)]
pub const M: u64 = 1_000_000_007;

fn main() {
    input! {
        n: u64, m: u64, k: u64
    }

    let cb = comb(n*m-2,k-2,M);
    let mut xlen = 0u64;
    for d in 0..n {
        let ptn = (n - d) * m * m % M;
        xlen += d * ptn % M * cb % M;
        xlen %= M;
    }
    let mut ylen = 0u64;
    for d in 0..m {
        let ptn = (m - d) * n * n % M;
        ylen += d * ptn % M * cb % M;
        ylen %= M;
    }
    println!("{}", (xlen + ylen) % M);
}

#[doc = " a / b modulo m == a * modinv(b, m) % m"]
#[allow(dead_code)]
fn modinv(a_: u64, m_: u64) -> u64 {
    let m = m_ as i64;
    let mut a = a_ as i64;
    let mut b = m as i64;
    let mut u = 1;
    let mut v = 0;
    while b != 0 {
        let t = a / b;
        a -= t * b;
        std::mem::swap(&mut a, &mut b);
        u -= t * v;
        std::mem::swap(&mut u, &mut v);
    }
    u %= m;
    if u < 0 {
        u += m
    };
    u as u64
}
#[allow(dead_code)]
fn comb(n: u64, r: u64, m: u64) -> u64 {
    let mut ret = 1u64;
    for i in 0..r {
        ret *= n - i;
        ret %= m;
    }
    for i in 1..r + 1 {
        ret *= modinv(i, m);
        ret %= m;
    }
    ret
}
#[doc = " combination with repetition"]
#[allow(dead_code)]
fn comb_rep(n: u64, r: u64, m: u64) -> u64 {
    comb(n + r - 1, r, m)
}