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 C# (Mono 4.6.2.0) (judge verdict: Accepted)

using Yorukatsu056.Questions;
using Yorukatsu056.Extensions;
using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using System.Text;

namespace Yorukatsu056.Questions
{
    public class QuestionF : AtCoderQuestionBase
    {
        public override IEnumerable<object> Solve(TextReader inputStream)
        {
            var nmk = inputStream.ReadIntArray();
            var height = nmk[0];
            var width = nmk[1];
            var k = nmk[2];

            yield return (Count(height, width, k) + Count(width, height, k)).Value;
        }

        Modular Count(int height, int width, int k)
        {
            var patterns = Modular.Combination(height * width - 2, k - 2);

            var sum = new Modular(0);
            for (int d = 1; d < width; d++)
            {
                sum += new Modular(d) * new Modular(height) * new Modular(height) * new Modular(width - d);
            }

            return patterns * sum;
        }

        [System.Runtime.InteropServices.StructLayout(System.Runtime.InteropServices.LayoutKind.Auto)]
        public struct Modular : IEquatable<Modular>, IComparable<Modular>
        {
            private const int _defaultMod = 1000000007;
            private readonly long _value;    // 0 <= value < Mod の値をとる
            public int Mod { get; }
            public int Value => (int)_value;

            public Modular(long value, int mod = _defaultMod)
            {
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }
                Mod = mod;

                if (value >= 0 && value < mod)
                {
                    _value = value;
                }
                else
                {
                    _value = value % mod;
                    if (Value < 0)
                    {
                        _value += mod;
                    }
                }
            }

            private static void CheckModEquals(Modular a, Modular b)
            {
                if (a.Mod != b.Mod)
                {
                    throw new ArgumentException($"{nameof(a)}, {nameof(b)}", $"両者の法（{nameof(Mod)}）は等しくなければなりません。");
                }
            }

            public static Modular operator +(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value + b._value;
                if (result > a.Mod)
                {
                    result -= a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator -(Modular a, Modular b)
            {
                CheckModEquals(a, b);

                var result = a._value - b._value;
                if (result < 0)
                {
                    result += a.Mod;    // 剰余演算を避ける
                }
                return new Modular(result, a.Mod);
            }

            public static Modular operator *(Modular a, Modular b)
            {
                CheckModEquals(a, b);
                return new Modular(a._value * b._value, a.Mod);
            }

            public static Modular operator /(Modular a, Modular b)  // こいつは値渡しの方が速いっぽい
            {
                CheckModEquals(a, b);
                return a * Pow(b, a.Mod - 2);
            }

            // 需要は不明だけど一応
            public static bool operator ==(Modular left, Modular right) => left.Equals(right);
            public static bool operator !=(Modular left, Modular right) => !(left == right);
            public static bool operator <(Modular left, Modular right) => left.CompareTo(right) < 0;
            public static bool operator <=(Modular left, Modular right) => left.CompareTo(right) <= 0;
            public static bool operator >(Modular left, Modular right) => left.CompareTo(right) > 0;
            public static bool operator >=(Modular left, Modular right) => left.CompareTo(right) >= 0;

            public static explicit operator int(Modular a) => a.Value;
            public static explicit operator long(Modular a) => a._value;

            public static Modular Pow(Modular a, int n)
            {
                switch (n)
                {
                    case 0:
                        return new Modular(1, a.Mod);
                    case 1:
                        return a;
                    default:
                        var p = Pow(a, n / 2);
                        return p * p * Pow(a, n % 2);
                }
            }

            private static Dictionary<int, List<int>> FactorialCache { get; } = new Dictionary<int, List<int>>();
            private static Dictionary<int, int[]> FactorialInverseCache { get; } = new Dictionary<int, int[]>();
            private static Dictionary<int, List<int>> PermutationCache { get; } = new Dictionary<int, List<int>>();
            const int maxFactorial = 1000000;

            public static Modular Factorial(int n, int mod = _defaultMod)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません。");
                }
                if (mod < 2)
                {
                    throw new ArgumentOutOfRangeException(nameof(mod), $"{nameof(mod)}は2以上の素数でなければなりません。");
                }

                if (!FactorialCache.ContainsKey(mod))
                {
                    FactorialCache.Add(mod, new List<int>() { 1 });
                }

                var cache = FactorialCache[mod];
                for (int i = cache.Count; i <= n; i++)  // Countが1（0!までキャッシュ済み）のとき1!～n!まで計算
                {
                    cache.Add((int)((long)cache[i - 1] * i % mod));
                }
                return new Modular(cache[n], mod);
            }

            public static Modular Permutation(int n, int r, int mod = _defaultMod)
            {
                CheckNR(n, r);

                if (!PermutationCache.ContainsKey(mod ^ n))
                {
                    PermutationCache.Add(mod ^ n, new List<int>() { 1 });
                }

                var cache = PermutationCache[mod ^ n];
                for (int i = cache.Count; i <= r; i++)  // Countが1（nP0までキャッシュ済み）のときnP1～nPrまで計算
                {
                    cache.Add((int)((long)cache[i - 1] * (n - (i - 1)) % mod));
                }
                return new Modular(cache[r], mod);
            }

            public static Modular Combination(int n, int r, int mod = _defaultMod)
            {
                if (!FactorialInverseCache.ContainsKey(mod))
                {
                    InitializeCombinationTable(maxFactorial, mod);
                }
                CheckNR(n, r);
                r = Math.Min(r, n - r);
                return n == r ? new Modular(1, mod) : new Modular(FactorialCache[mod][n], mod) * new Modular(FactorialInverseCache[mod][r], mod) * new Modular(FactorialInverseCache[mod][n - r], mod);
            }

            private static void InitializeCombinationTable(int max, int mod)
            {
                Factorial(max);
                FactorialInverseCache.Add(mod, new int[max + 1]);

                long fInv = (new Modular(1, mod) / Factorial(max, mod)).Value;
                FactorialInverseCache[mod][max] = (int)fInv;
                for (int i = max - 1; i >= 0; i--)
                {
                    fInv = (fInv * (i + 1)) % mod;
                    FactorialInverseCache[mod][i] = (int)fInv;
                }
            }

            public static Modular CombinationWithRepetition(int n, int r, int mod = _defaultMod) => Combination(n + r - 1, r, mod);

            private static void CheckNR(int n, int r)
            {
                if (n < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(n), $"{nameof(n)}は0以上の整数でなければなりません");
                }
                if (r < 0)
                {
                    throw new ArgumentOutOfRangeException(nameof(r), $"{nameof(r)}は0以上の整数でなければなりません。");
                }
                if (n < r)
                {
                    throw new ArgumentOutOfRangeException($"{nameof(n)},{nameof(r)}", $"{nameof(r)}は{nameof(n)}以下でなければなりません。");
                }
            }

            public override string ToString() => $"{_value} (mod {Mod})";

            public bool Equals(Modular other) => _value == other._value && Mod == other.Mod;

            public int CompareTo(Modular other) => Mod == other.Mod ? _value.CompareTo(other._value) : 0;

            public override int GetHashCode() => Value ^ Mod;
        }

    }
}

namespace Yorukatsu056
{
    class Program
    {
        static void Main(string[] args)
        {
            IAtCoderQuestion question = new QuestionF();    // 問題に合わせて書き換え
            var answers = question.Solve(Console.In);
            foreach (var answer in answers)
            {
                Console.WriteLine(answer);
            }
        }
    }
}

#region Base Classes

namespace Yorukatsu056.Questions
{

    public interface IAtCoderQuestion
    {
        IEnumerable<object> Solve(string input);
        IEnumerable<object> Solve(TextReader inputStream);
    }

    public abstract class AtCoderQuestionBase : IAtCoderQuestion
    {
        public IEnumerable<object> Solve(string input)
        {
            var stream = new MemoryStream(Encoding.Unicode.GetBytes(input));
            var reader = new StreamReader(stream, Encoding.Unicode);

            return Solve(reader);
        }

        public abstract IEnumerable<object> Solve(TextReader inputStream);
    }

}

#endregion

#region Extensions

namespace Yorukatsu056.Extensions
{
    internal static class TextReaderExtensions
    {
        internal static int ReadInt(this TextReader reader) => int.Parse(ReadString(reader));
        internal static long ReadLong(this TextReader reader) => long.Parse(ReadString(reader));
        internal static double ReadDouble(this TextReader reader) => double.Parse(ReadString(reader));
        internal static string ReadString(this TextReader reader) => reader.ReadLine();

        internal static int[] ReadIntArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(int.Parse).ToArray();
        internal static long[] ReadLongArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(long.Parse).ToArray();
        internal static double[] ReadDoubleArray(this TextReader reader, char separator = ' ') => ReadStringArray(reader, separator).Select(double.Parse).ToArray();
        internal static string[] ReadStringArray(this TextReader reader, char separator = ' ') => reader.ReadLine().Split(separator);
    }
}

#endregion