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 ( 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" ) } ; }

fn main() {
    input! {
        n: i64, m: i64, k: i64
    }

    let mut ret = Co(0);
    for d in 1..n {
        ret += Co(n-d) * Co(m) * Co(m) * Co(d);
    }
    for d in 1..m {
        ret += Co(m-d) * Co(n) * Co(n) * Co(d);
    }
    ret *= comb( (n*m-2) as u64, (k-2) as u64);
    println!("{}", ret);
}

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

#[allow(dead_code)]
fn comb(n: u64, r: u64) -> Co {
    let mut ret: Co = Co(1);
    for i in 0..r {
        ret *= (n - i).into();
    }
    for i in 1..r + 1 {
        ret /= i.into();
    }
    ret
}

#[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)
    }
}