Language: C++

#include <cassert>
#include <cstdio>
#include <cmath>
#include <iostream>
#include <iomanip>
#include <sstream>
#include <vector>
#include <set>
#include <map>
#include <queue>
#include <numeric>
#include <algorithm>

using namespace std;
using lint = long long;
constexpr int MOD = 1000000007, INF = 1010101010;
constexpr lint LINF = 1LL << 60;

template <class T>
ostream &operator<<(ostream &os, const vector<T> &vec) {
	for (const auto &e : vec) os << e << (&e == &vec.back() ? "" : " ");
	return os;
}

#ifdef _DEBUG
template <class T>
void dump(const char* str, T &&h) { cerr << str << " = " << h << "\n"; };
template <class Head, class... Tail>
void dump(const char* str, Head &&h, Tail &&... t) {
	while (*str != ',') cerr << *str++; cerr << " = " << h << "\n";
	dump(str + (*(str + 1) == ' ' ? 2 : 1), t...);
}
#define DMP(...) dump(#__VA_ARGS__, __VA_ARGS__)
#else 
#define DMP(...) ((void)0)
#endif

template<int Modulo = MOD> struct Mint {

	lint val;
	constexpr Mint(lint v = 0) noexcept : val(v % Modulo) { if (val < 0) v += Modulo; }

	constexpr Mint& operator += (const Mint &r) noexcept {
		val += r.val;
		if (val >= Modulo) val -= Modulo;
		return *this;
	}
	constexpr Mint& operator -= (const Mint &r) noexcept {
		val -= r.val;
		if (val < 0) val += Modulo;
		return *this;
	}
	constexpr Mint& operator *= (const Mint &r) noexcept {
		val = val * r.val % Modulo;
		return *this;
	}
	constexpr Mint& operator /= (const Mint &r) noexcept {
		lint a = r.val, b = Modulo, u = 1, v = 0;
		while (b) {
			lint t = a / b;
			a -= t * b; swap(a, b);
			u -= t * v; swap(u, v);
		}
		val = val * u % Modulo;
		if (val < 0) val += Modulo;
		return *this;
	}

	constexpr Mint operator + (const Mint &r) const noexcept { return Mint(*this) += r; }
	constexpr Mint operator - (const Mint &r) const noexcept { return Mint(*this) -= r; }
	constexpr Mint operator * (const Mint &r) const noexcept { return Mint(*this) *= r; }
	constexpr Mint operator / (const Mint &r) const noexcept { return Mint(*this) /= r; }

	constexpr int getmod() { return Modulo; }
	constexpr Mint operator - () const noexcept { return val ? Modulo - val : 0; }

	constexpr bool operator == (const Mint &r) const noexcept { return val == r.val; }
	constexpr bool operator != (const Mint &r) const noexcept { return val != r.val; }

	friend ostream &operator << (ostream &os, const Mint<Modulo> &x) noexcept { return os << x.val; }
	friend istream &operator >> (istream &is, Mint<Modulo> &x) noexcept {
		lint tmp; is >> tmp;
		x = Mint<Modulo>(tmp);
		return is;
	}
	friend constexpr Mint<Modulo> modpow(const Mint<Modulo> &a, lint n) noexcept {
		Mint res(1), tmp = a;
		while (n > 0) {
			if (n & 1) res *= tmp;
			tmp *= tmp;
			n >>= 1;
		}
		return res;
	}
};

using mint = Mint<>;

class Factorial {
private:
	vector<mint> fac, inv, finv;

	void init(int N) {
		fac[0] = fac[1] = 1; inv[1] = 1; finv[0] = finv[1] = 1;

		for (int i = 2; i < N; i++) {
			fac[i] = fac[i - 1] * i;
			inv[i] = -inv[MOD%i] * (MOD / i);
			finv[i] = finv[i - 1] * inv[i];
		}
	}

public:
	Factorial(int N = 110000) : fac(N + 1), inv(N + 1), finv(N + 1) { init(N + 1); }

	mint operator() (int n, int k) {
		if (n < k || k < 0) return 0LL;
		return fac[n] * (finv[k] * finv[n - k]);
	}

	mint Pmod(int n, int k) {
		if (n < k || k < 0) return 0LL;
		return fac[n] * finv[n - k];
	}
};

int main() {

	cin.tie(nullptr);
	ios::sync_with_stdio(false);

	Factorial combi(210000);

	mint n, m, k;
	cin >> n >> m >> k;

	mint ans = 0;
	for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
	for (int d = 1; d < n; d++) ans += d * (n - d) * m*m;
	cout << ans * combi(n*m - 2, k - 2) << "\n";

	return 0;
}

Compiler output:
a.cc: In function 'int main()':
a.cc:138:27: error: no match for 'operator<' (operand types are 'int' and 'mint' {aka 'Mint<>'})
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                         ~ ^ ~
      |                         |   |
      |                         int mint {aka Mint<>}
In file included from /usr/include/c++/14/bits/stl_algobase.h:64,
                 from /usr/include/c++/14/bits/specfun.h:43,
                 from /usr/include/c++/14/cmath:3906,
                 from a.cc:3:
/usr/include/c++/14/bits/stl_pair.h:1045:5: note: candidate: 'template<class _T1, class _T2> constexpr bool std::operator<(const pair<_T1, _T2>&, const pair<_T1, _T2>&)'
 1045 |     operator<(const pair<_T1, _T2>& __x, const pair<_T1, _T2>& __y)
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_pair.h:1045:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::pair<_T1, _T2>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
In file included from /usr/include/c++/14/bits/stl_algobase.h:67:
/usr/include/c++/14/bits/stl_iterator.h:448:5: note: candidate: 'template<class _Iterator> constexpr bool std::operator<(const reverse_iterator<_Iterator>&, const reverse_iterator<_Iterator>&)'
  448 |     operator<(const reverse_iterator<_Iterator>& __x,
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_iterator.h:448:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::reverse_iterator<_Iterator>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/bits/stl_iterator.h:493:5: note: candidate: 'template<class _IteratorL, class _IteratorR> constexpr bool std::operator<(const reverse_iterator<_Iterator>&, const reverse_iterator<_IteratorR>&)'
  493 |     operator<(const reverse_iterator<_IteratorL>& __x,
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_iterator.h:493:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::reverse_iterator<_Iterator>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/bits/stl_iterator.h:1694:5: note: candidate: 'template<class _IteratorL, class _IteratorR> constexpr bool std::operator<(const move_iterator<_IteratorL>&, const move_iterator<_IteratorR>&)'
 1694 |     operator<(const move_iterator<_IteratorL>& __x,
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_iterator.h:1694:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::move_iterator<_IteratorL>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/bits/stl_iterator.h:1760:5: note: candidate: 'template<class _Iterator> constexpr bool std::operator<(const move_iterator<_IteratorL>&, const move_iterator<_IteratorL>&)'
 1760 |     operator<(const move_iterator<_Iterator>& __x,
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_iterator.h:1760:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::move_iterator<_IteratorL>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
In file included from /usr/include/c++/14/bits/basic_string.h:47,
                 from /usr/include/c++/14/string:54,
                 from /usr/include/c++/14/bits/locale_classes.h:40,
                 from /usr/include/c++/14/bits/ios_base.h:41,
                 from /usr/include/c++/14/ios:44,
                 from /usr/include/c++/14/ostream:40,
                 from /usr/include/c++/14/iostream:41,
                 from a.cc:4:
/usr/include/c++/14/string_view:673:5: note: candidate: 'template<class _CharT, class _Traits> constexpr bool std::operator<(basic_string_view<_CharT, _Traits>, basic_string_view<_CharT, _Traits>)'
  673 |     operator< (basic_string_view<_CharT, _Traits> __x,
      |     ^~~~~~~~
/usr/include/c++/14/string_view:673:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'std::basic_string_view<_CharT, _Traits>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/string_view:680:5: note: candidate: 'template<class _CharT, class _Traits> constexpr bool std::operator<(basic_string_view<_CharT, _Traits>, __type_identity_t<basic_string_view<_CharT, _Traits> >)'
  680 |     operator< (basic_string_view<_CharT, _Traits> __x,
      |     ^~~~~~~~
/usr/include/c++/14/string_view:680:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'std::basic_string_view<_CharT, _Traits>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/string_view:688:5: note: candidate: 'template<class _CharT, class _Traits> constexpr bool std::operator<(__type_identity_t<basic_string_view<_CharT, _Traits> >, basic_string_view<_CharT, _Traits>)'
  688 |     operator< (__type_identity_t<basic_string_view<_CharT, _Traits>> __x,
      |     ^~~~~~~~
/usr/include/c++/14/string_view:688:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   'Mint<>' is not derived from 'std::basic_string_view<_CharT, _Traits>'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/bits/basic_string.h:3874:5: note: candidate: 'template<class _CharT, class _Traits, class _Alloc> bool std::operator<(const __cxx11::basic_string<_CharT, _Traits, _Allocator>&, const __cxx11::basic_string<_CharT, _Traits, _Allocator>&)'
 3874 |     operator<(const basic_string<_CharT, _Traits, _Alloc>& __lhs,
      |     ^~~~~~~~
/usr/include/c++/14/bits/basic_string.h:3874:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::__cxx11::basic_string<_CharT, _Traits, _Allocator>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/bits/basic_string.h:3888:5: note: candidate: 'template<class _CharT, class _Traits, class _Alloc> bool std::operator<(const __cxx11::basic_string<_CharT, _Traits, _Allocator>&, const _CharT*)'
 3888 |     operator<(const basic_string<_CharT, _Traits, _Alloc>& __lhs,
      |     ^~~~~~~~
/usr/include/c++/14/bits/basic_string.h:3888:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::__cxx11::basic_string<_CharT, _Traits, _Allocator>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
/usr/include/c++/14/bits/basic_string.h:3901:5: note: candidate: 'template<class _CharT, class _Traits, class _Alloc> bool std::operator<(const _CharT*, const __cxx11::basic_string<_CharT, _Traits, _Allocator>&)'
 3901 |     operator<(const _CharT* __lhs,
      |     ^~~~~~~~
/usr/include/c++/14/bits/basic_string.h:3901:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const _CharT*' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
In file included from /usr/include/c++/14/bits/memory_resource.h:47,
                 from /usr/include/c++/14/string:68:
/usr/include/c++/14/tuple:2600:5: note: candidate: 'template<class ... _TElements, class ... _UElements> constexpr bool std::operator<(const tuple<_UTypes ...>&, const tuple<_Elements ...>&)'
 2600 |     operator<(const tuple<_TElements...>& __t,
      |     ^~~~~~~~
/usr/include/c++/14/tuple:2600:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::tuple<_UTypes ...>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
In file included from /usr/include/c++/14/vector:66,
                 from a.cc:7:
/usr/include/c++/14/bits/stl_vector.h:2089:5: note: candidate: 'template<class _Tp, class _Alloc> bool std::operator<(const vector<_Tp, _Alloc>&, const vector<_Tp, _Alloc>&)'
 2089 |     operator<(const vector<_Tp, _Alloc>& __x, const vector<_Tp, _Alloc>& __y)
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_vector.h:2089:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::vector<_Tp, _Alloc>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
In file included from /usr/include/c++/14/set:63,
                 from a.cc:8:
/usr/include/c++/14/bits/stl_set.h:1025:5: note: candidate: 'template<class _Key, class _Compare, class _Alloc> bool std::operator<(const set<_Key, _Compare, _Allocator>&, const set<_Key, _Compare, _Allocator>&)'
 1025 |     operator<(const set<_Key, _Compare, _Alloc>& __x,
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_set.h:1025:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::set<_Key, _Compare, _Allocator>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |                             ^
In file included from /usr/include/c++/14/set:64:
/usr/include/c++/14/bits/stl_multiset.h:1011:5: note: candidate: 'template<class _Key, class _Compare, class _Alloc> bool std::operator<(const multiset<_Key, _Compare, _Allocator>&, const multiset<_Key, _Compare, _Allocator>&)'
 1011 |     operator<(const multiset<_Key, _Compare, _Alloc>& __x,
      |     ^~~~~~~~
/usr/include/c++/14/bits/stl_multiset.h:1011:5: note:   template argument deduction/substitution failed:
a.cc:138:29: note:   mismatched types 'const std::multiset<_Key, _Compare, _Allocator>' and 'int'
  138 |         for (int d = 1; d < m; d++) ans += d * (m - d) * n*n;
      |