Problem:
Score : 900 points

Problem Statement

There is a rectangular sheet of paper with height H+1 and width W+1. We introduce an xy-coordinate system so that the four corners of the sheet are (0, 0), (W + 1, 0), (0, H + 1) and (W + 1, H + 1).

This sheet can be cut along the lines x = 1,2,...,W and the lines y = 1,2,...,H. Consider a sequence of operations of length K where we choose K of these H + W lines and cut the sheet along those lines one by one in some order.

Let the score of a cut be the number of pieces of paper that exist just after the cut. The score of a sequence of operations is the sum of the scores of all of the K cuts.

Find the sum of the scores of all possible sequences of operations of length K. Since this value can be extremely large, print the number modulo 10^9 + 7.

Constraints

1 \leq H,W \leq 10^7

1 \leq K \leq H + W

H, W and K are integers.

Input

Input is given from Standard Input in the following format:

H W K

Output

Print the sum of the scores, modulo 10^9 + 7.

Sample Input 1

2 1 2

Sample Output 1

34

Let x_1, y_1 and y_2 denote the cuts along the lines x = 1, y = 1 and y = 2, respectively. The six possible sequences of operations and the score of each of them are as follows:

y_1, y_2: 2 + 3 = 5

y_2, y_1: 2 + 3 = 5

y_1, x_1: 2 + 4 = 6

y_2, x_1: 2 + 4 = 6

x_1, y_1: 2 + 4 = 6

x_1, y_2: 2 + 4 = 6

The sum of these is 34.

Sample Input 2

30 40 50

Sample Output 2

616365902

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

Solution in Rust (1.15.1) (judge verdict: Accepted)

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)*}
    };
}

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)*}
    };
}

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")
    };
}

fn main() {
    input!{
        h: usize,
        w: usize,
        k: usize
    }
    let m = 1000000007;
    let mut ans = k * (k - 1) % m * (k + 1) % m * stair_m(h + w - 2, m) % m * inv_m(3, m) % m * h % m * w % m;
    ans = (ans + k * (k + 3) / 2 % m * stair_m(h + w, m) % m) % m;
    ans = ans * inv_m(stair_m(h + w - k, m), m) % m;
    println!("{}", ans);
}

fn stair_m(n: usize, d: usize) -> usize {
    (1..n+1).fold(1, |ac, x| ac * x % d)
}

fn pow_m(n: usize, mut p: usize, d: usize) -> usize {
    let mut r = n;
    let mut ret = 1;
    while p > 0 {
        if p % 2 == 0 {
            r = r * r % d;
            p /= 2;
        } else {
            ret = ret * r % d;
            p -= 1;
        }
    }
    ret
}

// only when d is prime
fn inv_m(n: usize, d: usize) -> usize {
    pow_m(n, d - 2, d)
}