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.Text;

class Util{
	static void Main(){
		Sol mySol =new Sol();
		mySol.Solve();
	}
}

class Sol{
	public void Solve(){
		int[] HWK = ria();

        List<char[]> cs = new List<char[]>();
        int blackCnt = 0;
        int[] blackCnt_forH = new int[HWK[0]];
        int[] blackCnt_forW = new int[HWK[1]];
        for(int i = 0; i < HWK[0]; i++) {
            char[] chars = new char[HWK[1]];
            string str = rs();

            for(int j = 0; j < HWK[1]; j++) {
                if(str[j] == '#') {
                    blackCnt++;
                    blackCnt_forH[i]++;
                    blackCnt_forW[j]++;
                }
                chars[j] = str[j];
            }
            cs.Add(chars);
        }

        int ans = 0;
        for(int i = 0; i < (1 << HWK[0]); i++) {
            int[] cond_H = new int[HWK[0]];
            int blackCnt_current_H = blackCnt;
            int drawCnt = 0;

            for(int j = 0; j < HWK[0]; j++) {
                cond_H[j] = 1 & (i >> j);

                if(cond_H[j] == 1) {
                    blackCnt_current_H -= blackCnt_forH[j];
                    drawCnt++;
                }
            }

            for(int i2 = 0; i2 < (1 << HWK[1]); i2++) {
                int[] cond_W = new int[HWK[1]];
                int blackCnt_current_W = blackCnt_current_H;

                for(int j2 = 0; j2 < HWK[1]; j2++) {
                    cond_W[j2] = 1 & (i2 >> j2);

                    if(cond_W[j2] == 1) {

                        for(int k = 0; k < HWK[0]; k++) {
                            if(cond_H[k] == 1) continue;

                            if(cs[k][j2] == '#') blackCnt_current_W--;
                        }
                    }
                }
                
                if(blackCnt_current_W == HWK[2]) ans++;
            }
        }

        // List<int> blackCnt_H = new List<int>();
        // // List<int[]> pattern_H = new List<int[]>();
        // for(int i = 0; i < (1 << HWK[0]); i++) {
        //     int[] cond = new int[HWK[0]];
        //     int blackCnt_current = blackCnt;
        //     bool isOver = false;

        //     for(int j = 0; j < HWK[0]; j++) {
        //         cond[j] = 1 & (i >> j);

        //         if(cond[j] == 1) {
        //             blackCnt_current -= blackCnt_forH[j];
                    
        //             if(blackCnt_current < HWK[2]) {
        //                 isOver = true;
        //                 break;
        //             }
        //         }
        //     }
        //     if(isOver) continue;

        //     blackCnt_H.Add(blackCnt_current);
        //     // pattern_H.Add(cond);
        // }
        // blackCnt_H.Sort();

        // List<int> blackCnt_W = new List<int>();
        // // List<int[]> pattern_W = new List<int[]>();
        // for(int i = 0; i < (1 << HWK[1]); i++) {
        //     int[] cond = new int[HWK[1]];
        //     int blackCnt_current = blackCnt;
        //     bool isOver = false;

        //     for(int j = 0; j < HWK[1]; j++) {
        //         cond[j] = 1 & (i >> j);

        //         if(cond[j] == 1) {
        //             blackCnt_current -= blackCnt_forW[j];
                    
        //             if(blackCnt_current < HWK[2]) {
        //                 isOver = true;
        //                 break;
        //             }
        //         }
        //     }
        //     if(isOver) continue;

        //     blackCnt_W.Add(blackCnt_current);
        //     // pattern_W.Add(cond);
        // }
        // blackCnt_W.Sort();
        
		Console.WriteLine(ans);
		Console.ReadLine();
	}

	static String rs(){return Console.ReadLine();}
	static int ri(){return int.Parse(Console.ReadLine());}
	static long rl(){return long.Parse(Console.ReadLine());}
	static double rd(){return double.Parse(Console.ReadLine());}
	static String[] rsa(char sep=' '){return Console.ReadLine().Split(sep);}
	static int[] ria(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>int.Parse(e));}
	static long[] rla(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>long.Parse(e));}
	static double[] rda(char sep=' '){return Array.ConvertAll(Console.ReadLine().Split(sep),e=>double.Parse(e));}
    static int[] generateNums(int num, int N){return Enumerable.Repeat(num, N).ToArray();}
    static int[] generateSuretsu(int N){return Enumerable.Range(0, N).ToArray();}
}

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();
            }
        }
    }
}