Problem:
Score : 300 points

Problem Statement

We have a grid of H rows and W columns of squares. The color of the square at the i-th row from the top and the j-th column from the left (1 \leq i \leq H, 1 \leq j \leq W) is given to you as a character c_{i,j}: the square is white if c_{i,j} is ., and black if c_{i,j} is #.

Consider doing the following operation:

Choose some number of rows (possibly zero), and some number of columns (possibly zero). Then, paint red all squares in the chosen rows and all squares in the chosen columns.

You are given a positive integer K. How many choices of rows and columns result in exactly K black squares remaining after the operation? Here, we consider two choices different when there is a row or column chosen in only one of those choices.

Constraints

1 \leq H, W \leq 6

1 \leq K \leq HW

c_{i,j} is . or #.

Input

Input is given from Standard Input in the following format:

H W K
c_{1,1}c_{1,2}...c_{1,W}
c_{2,1}c_{2,2}...c_{2,W}
:
c_{H,1}c_{H,2}...c_{H,W}

Output

Print an integer representing the number of choices of rows and columns satisfying the condition.

Sample Input 1

2 3 2
..#
###

Sample Output 1

5

Five choices below satisfy the condition.

The 1-st row and 1-st column

The 1-st row and 2-nd column

The 1-st row and 3-rd column

The 1-st and 2-nd column

The 3-rd column

Sample Input 2

2 3 4
..#
###

Sample Output 2

1

One choice, which is choosing nothing, satisfies the condition.

Sample Input 3

2 2 3
##
##

Sample Output 3

0

Sample Input 4

6 6 8
..##..
.#..#.
#....#
######
#....#
#....#

Sample Output 4

208

Solution in C# (.NET Core 3.1.201) (judge verdict: Accepted)

using System;
using System.Collections;
using System.Collections.Generic;
using System.Linq;
using System.IO;

public class MainClass
{
    public static void Main(string[] args)
    {
        var sw = new StreamWriter(Console.OpenStandardOutput(), Console.Out.Encoding);
        sw.AutoFlush = false;
        Console.SetOut(sw);

        if (args.Length > 0 && args[0] == "debug")
        {
            var inputs = new[] { @"
2 3 2
..#
###
				", @"
2 3 4
..#
###
				", @"
2 2 3
##
##
				", @"
6 6 8
..##..
.#..#.
#....#
######
#....#
#....#
				", @"

                ", @"

				", @"

				", @"

				" };

            foreach (var str in inputs)
            {
                var text = str.TrimStart();
                if (text.Length == 0) continue;
                var reader = new System.IO.StringReader(text);
                Console.SetIn(reader);
                Main2(args);
                Console.WriteLine("----------------------------------");
            }
        }
        else
            Main2(args);

        Console.Out.Flush();
    }

    private static void Main2(string[] args)
    {
        checked
        {
            var /*HWK*/ (H, W, K) = (Int(), Int(), Int());
            var C = EnumerablePlus.Repeat(H, Console.ReadLine).ToArray();
            var res = 0;

            for (int hMask = 0, hMax = (1 << H) - 1; hMask <= hMax; hMask++)
            {
                for (int wMask = 0, wMax = (1 << W) - 1; wMask <= wMax; wMask++)
                {
                    int k = 0;
                    for (int h = 0; h < H; h++)
                    {
                        if ((hMask & (1 << h)) != 0) continue;

                        for (int w = 0; w < W; w++)
                        {
                            if ((wMask & (1 << w)) != 0) continue;

                            if (C[h][w] == '#') k++;
                        }
                    }

                    if (k == K) res++;
                }
            }

            Console.WriteLine(res);
        }
    }

    #region "入力"

    private static string NextInput = "";

    private static int Int() => int.Parse(Str());
    private static long Long() => long.Parse(Str());
    private static float Float() => float.Parse(Str());
    private static double Double() => double.Parse(Str());
    private static string Str()
    {
        var s = NextInput;
        if (s == "") s = Console.ReadLine();

        var idx = s.IndexOf(' ');
        if (idx >= 0)
        {
            NextInput = s.Substring(idx + 1);
            return s.Substring(0, idx);
        }
        NextInput = "";
        return s;
    }

    #endregion

}

public static class EnumerablePlus
{
    ///<summary>出力文字列に変換。</summary>
    public static string MkString<T>(this IEnumerable<T> list, string sep = null)
    {
        if (sep == null) sep = " ";

        return list.Aggregate(
            new System.Text.StringBuilder(),
            (a, b) => ((a.Length > 0) ? a.Append(sep) : a).Append(b)
            ).ToString();
    }

    ///<summary>添字付きfor。</summary>
    public static void ForEachWithIdx<T>(this IEnumerable<T> list, Action<T, long> a)
    {
        long i = 0;
        foreach (var item in list)
        {
            a(item, i++);
        }
    }

    ///<summary>添字なしfor。</summary>
    public static void ForEach<T>(this IEnumerable<T> list, Action<T> a)
    {
        foreach (var item in list)
        {
            a(item);
        }
    }

    ///<summary>指定回数デリゲートを呼び出す。</summary>
    public static IEnumerable<T> Repeat<T>(long n, Func<T> f)
    {
        for (long i = 0; i < n; i++)
        {
            yield return f();
        }
    }

    ///<summary>条件付きMin。</summary>
    public static T MinBy<T, U>(this IEnumerable<T> xs, Func<T, U> key) where U : IComparable<U>
    {
        return xs.Aggregate((a, b) => key(a).CompareTo(key(b)) <= 0 ? a : b);
    }

    ///<summary>条件付きMax。</summary>
    public static T MaxBy<T, U>(this IEnumerable<T> xs, Func<T, U> key) where U : IComparable<U>
    {
        return xs.Aggregate((a, b) => key(a).CompareTo(key(b)) >= 0 ? a : b);
    }

    ///<summary>条件付きMax。複数の値を返す。</summary>
    public static List<T> MaxByList<T, U>(this IEnumerable<T> xs, Func<T, U> key) where U : IComparable<U>
    {
        return xs.Aggregate(
            new List<T>(),
            (list, b) =>
            {
                if (list.Count == 0) { list.Add(b); return list; }

                var aa = key(list[0]);
                var bb = key(b);

                if (aa.CompareTo(bb) < 0) list.Clear();

                if (aa.CompareTo(bb) <= 0) { list.Add(b); return list; }

                return list;
            });
    }

    ///<summary>複数の値を返す。</summary>
    public static List<T> MaxList<T>(this IEnumerable<T> xs) where T : IComparable<T>
    {
        return xs.MaxByList(o => o);
    }
}

public static class MathPlus
{
    ///<summary>n(>=0)をradix進数に変換する。桁の小さい順に返す。</summary>
    public static List<long> NRadix(long n, long radix)
    {
        if (n < 0) throw new ArgumentOutOfRangeException(nameof(n));
        if (radix < 2) throw new ArgumentOutOfRangeException(nameof(n));

        var res = new List<long>();
        if (n == 0L)
        {
            res.Add(0);
            return res;
        }

        while (n > 0L)
        {
            res.Add(n % radix);
            n /= radix;
        }

        return res;
    }

    ///<summary>べき乗。高速な実装である。</summary>
    public static long Power(long x, long n)
    {
        if (n == 0) return 1;
        if (n % 2 == 0) return Power(x * x, n / 2);
        return x * Power(x, n - 1);
    }

    ///<summary>約数。順番は不定。</summary>
    public static List<long> Divisor(long n)
    {
        var list = Prime(n).
            GroupBy(id => id).
            Select(pair => new { prime = pair.Key, power = pair.Count() }).
            ToList();

        var result = list.Aggregate(new List<long>(), (left, pair) =>
        {
            var list2 = Enumerable.
                Range(0, pair.power + 1).
                Aggregate(new List<long>(), (res, l) =>
                {
                    var next = res.Count > 0 ? res.Last() * pair.prime : 1L;
                    res.Add(next);
                    return res;
                });

            if (left.Count == 0) return list2.ToList();

            return list2.
                SelectMany(l => left.Select(m => l * m)).
                ToList();
        });

        return result;
    }

    ///<summary>順列の数。</summary>
    public static long Permutation(long n, long r)
        => Factorial(n, r);

    ///<summary>組合せの数。</summary>
    public static long Combination(long n, long r)
    {
        if (r == 0) return 1;

        return Factorial(n, r) / Factorial(r);
    }

    ///<summary>階乗。カウントも指定可能。</summary>
    public static long Factorial(long a, long count = long.MaxValue)
    {
        long res = 1;

        for (long i = 0; a > 0 && i < count; i++)
        {
            checked { res *= a; }
            --a;
        }

        return res;
    }

    ///<summary>素因数分解。昇順。</summary>
    public static List<long> Prime(long a)
    {
        var res = new List<long>();

        var a2 = Math.Ceiling(Math.Sqrt(a));
        for (long b = 2; b <= a2; ++b)
        {
            while (a % b == 0)
            {
                res.Add(b);
                a = a / b;
                a2 = Math.Ceiling(Math.Sqrt(a));
            }
        }

        if (a > 1) res.Add(a);
        return res;
    }

    ///<summary>最小公倍数。</summary>
    public static long Lcm(IEnumerable<long> list)
    {
        return list.Aggregate((a, b) => Lcm(a, b));
    }

    ///<summary>最小公倍数。</summary>
    public static long Lcm(long a, long b)
    {
        return checked(a * b) / Gcd(a, b);
    }

    ///<summary>最大公約数。</summary>
    public static long Gcd(IEnumerable<long> list)
    {
        return list.Aggregate((a, b) => Gcd(a, b));
    }

    ///<summary>最大公約数。</summary>
    public static long Gcd(long a, long b)
    {
        if (a < b)
            Gcd(b, a);

        while (b != 0)
        {
            var rem = a % b;
            a = b;
            b = rem;
        }

        return a;
    }

    ///<summary>整数の切り上げ。</summary>
    public static long Ceiling(long a, long b) => (a + (b - 1)) / b;

    ///<summary>区間内の合計。beginも１から？end含む。</summary>
    public static ulong Sum(ulong begin, ulong end)
    {
        var up = Sum(end);
        var low = Sum(begin - 1);
        return up - low;
    }

    ///<summary>1からの合計。</summary>
    public static ulong Sum(ulong end) => checked((end + 1) * end / 2);
}

public static class Util
{
    ///<summary>リスト内の値を移動する。</summary>
    public static void Move<T>(this IList<T> list, int from, int to)
    {
        var item = list[from];

        for (var idx = from; idx > to; --idx)
        {
            list[idx] = list[idx - 1];
        }

        list[to] = item;
    }

    ///<summary>最後の値か、デフォルト値を取得。</summary>
    public static T LastOrValue<T>(this IList<T> list, T value)
    {
        return list.Count > 0 ? list.Last() : value;
    }

    ///<summary>辞書をMultiSetとして使うときのAdd。</summary>
    public static void AddMulti<T>(this IDictionary<T, int> xs, T key)
    {
        if (xs.ContainsKey(key))
            xs[key]++;
        else
            xs.Add(key, 1);
    }

    ///<summary>値を交換する。</summary>
    public static void Swap(ref int x, ref int y)
    {
        int temp = x;
        x = y;
        y = temp;
    }

    public static void Assert(bool cond)
    {
        if (!cond) throw new Exception();
    }
}