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;

namespace atcoder173
{
    class Program
    {
        static void Main(string[] args)
        {

            var list = Console.ReadLine().Split(' ').Select(x => int.Parse(x)).ToArray();
            var H = list[0]; // 行数
            var W = list[1]; // 列数
            var K = list[2]; // 期待する黒の個数

            string[,] field = new string[H, W];
            for (int i = 0; i < H; i++)
            {
                var buf = Console.ReadLine();
                for (int j = 0; j < W; j++){
                    field[i, j] = buf[j].ToString();
                }
            }

            int ans = 0;

            // rowBitCounter
            // どの行を選ぶか全ての組み合わせ数を定義する
            // 行を選ぶ選ばない含め全種類を試すため(1 << H)通り
            // 例えばHが2ならどの行も選ばない、0行を選ぶ、1行を選ぶ、0と1行を選ぶで4通り
            for (int rowBitCounter = 0; rowBitCounter < 1 << H; rowBitCounter++)
            {
                var selectedRowBitNums = new List<int>();

                // どの行を赤く塗りつぶすかを決めて、塗りつぶす行をselectedRowBitNumsに追加する
                // (1 << i)がBitでの行を表している
                // 例えばHが2なら(1 << i)が01,10の2通りでこれは0列目,1列目を表す(iがちょうど列番号になる)
                // rowBitCounterはどの行を選ぶかが決まるためrowBitCounterが0の時は下1桁は0でどの行も選ばない
                // rowBitCounterが1の時は下1桁が1なので、下1桁が1である01(0列目)が選ばれ、0列目(i)がselectedRowBitNumsに格納される
                for (int i = 0; i < H; i++){
                    if ((rowBitCounter & (1 << i)) != 0){
                        selectedRowBitNums.Add(i);                     
                    }
                }

                // columnBitCounter
                // どの列を選ぶか全ての組み合わせ数を定義する
                // 列を選ぶ選ばない含め全種類を試すため(1 << W)通り
                for (int columnBitCounter = 0; columnBitCounter < 1 << W; columnBitCounter++)
                {
                    var selectedColumnBitNums = new List<int>();

                    // どの列を赤く塗りつぶすかを決めて、塗りつぶす列をselectedRowBitNumsに追加する
                    // (1 << i)がBitでの列を表している
                    for (int i = 0; i < W; i++){
                        if ((columnBitCounter & (1 << i)) != 0){
                            selectedColumnBitNums.Add(i);
                        }
                    }

                    var fieldForCalculation = (string[,])field.Clone();

                    // 選択した行番号は赤で塗りつぶすためrで置き換える
                    foreach (var v in selectedRowBitNums){
                        for (int i = 0; i < W; i++){
                            fieldForCalculation[v, i] = "r";
                        }
                    }


                    foreach (var v in selectedColumnBitNums){
                        for (int i = 0; i < H; i++){
                            fieldForCalculation[i, v] = "r";
                        }
                    }
                    
                    int blackNum = 0;
                    for (int i = 0; i < fieldForCalculation.GetLength(0); i++){
                        for (int j = 0; j < fieldForCalculation.GetLength(1); j++){
                            if (fieldForCalculation[i, j] == "#") blackNum++;
                        }
                    }

                    if (blackNum == K) ans++;
                }

            }

            Console.WriteLine(ans);
        }
    }
}