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.Generic;
using System.Linq;

public class Hello
{
    public static void Main()
     {
        hoge();
        System.Console.ReadLine();
    }

    public static void hoge()
    {
        var vars = System.Console.ReadLine().Split(' ').Select(int.Parse).ToArray();

        var h = vars[0];
        var w = vars[1];
        var k = vars[2];

        var hCt = new List<List<int>>();

        for(int i = 0; i < h; i++)
        {
            var line = System.Console.ReadLine();
            hCt.Add(new List<int>());
            for(int j = 0; j < w; j++)
            {
                if(line[j] == '#')
                {
                    hCt[i].Add(j + 1);
                }
                
            }
        }
        var nums = Enumerable.Range(1, h).ToArray();

        List<int[]> lsth = new List<int[]>();

        for(int kumi = 1; kumi <=h; kumi++)
        {
            var combinations = Combination.Enumerate(nums, kumi, withRepetition: false);
            lsth.AddRange(combinations.ToArray());
        }

        nums = Enumerable.Range(1, w).ToArray();

        List<int[]> lstw = new List<int[]>();

        for (int kumi = 1; kumi <= w; kumi++)
        {
            var combinations = Combination.Enumerate(nums, kumi, withRepetition: false);
            lstw.AddRange(combinations.ToArray());
        }

        var ans = 0;

        for (int i = 0; i < lsth.Count; i++)
        {
            for(int j = 0; j < lstw.Count; j++)
            {
                var count = 0;
                var ct = new List<List<int>>();
                if(i != -1)
                {
                    foreach(var idx in lsth[i])
                    {
                        ct.Add(hCt[idx - 1]);
                    }
                }

                if(j != -1)
                {
                    ct.ForEach(lst => count += lst.Where(x => lstw[j].Contains(x)).Count());
                }

                if(count == k)
                {
                    ans++;
                }
            }
        }

        System.Console.WriteLine(ans);

    }

    public static class Combination
    {
        public static IEnumerable<T[]> Enumerate<T>(IEnumerable<T> items, int k, bool withRepetition)
        {
            if (k == 1)
            {
                foreach (var item in items)
                    yield return new T[] { item };
                yield break;
            }
            foreach (var item in items)
            {
                var leftside = new T[] { item };

                // item よりも前のものを除く （順列と組み合わせの違い)
                // 重複を許さないので、unusedから item そのものも取り除く
                var unused = withRepetition ? items : items.SkipWhile(e => !e.Equals(item)).Skip(1).ToList();

                foreach (var rightside in Enumerate(unused, k - 1, withRepetition))
                {
                    yield return leftside.Concat(rightside).ToArray();
                }
            }
        }
    }
}