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.Diagnostics;
using System.Linq;
using System.Text;
using System.Numerics;

namespace ABC173.C
{
	using static Util;
	using static Math;

	public class Solver : SolverBase
	{
		public void Run() {
			ReadInt3(out var H, out var W, out var K);
			List<string> lines = new List<string>();
			for (int i = 0; i < H; i++) {
				var line = ReadLine();
				lines.Add(line);
			}

			int ans = 0;

			// 残す行をBIT全探索
			// 全探索に使用する全bit数
			var n = H;

			// bit mask の生成
			// 2^n     = 1 << n
			for (ulong mask = 0; mask <= ((1ul << n) - 1); mask++) {

				var lineCnt = new int[W];

				// 各bitを下から舐める
				for (int i = 0; i < n; i++) {

					//iビット目が立っている？
					if (0 != (mask & (1ul << i))) {
						// 黒ますならインクリメント
						for (int j = 0; j < W; j++) {
							if (lines[i][j] == '#') {
								lineCnt[j]++;
							}
						}
					}
				}

				// さらにBIT全探索 = W方向
				var nw = W;

				for (ulong maskw = 0; maskw <= ((1ul << nw) - 1); maskw++) {

					var cnt = 0;

					// 各bitを下から舐める
					for (int i = 0; i < nw; i++) {

						//iビット目が立っている？
						if (0 != (maskw & (1ul << i))) {
							// 黒ますならインクリメント
							cnt += lineCnt[i];
						}
					}

					if (cnt == K) {
						++ans;
					}
				}
			}
			WriteLine(ans);
		}
#if !MYHOME
		static void Main(string[] args) {
			new Solver().Run();
		}
#endif
	}

	public static class Util
	{
		/// <summary>反転した新しいstringを返す</summary>
		public static string ReverseStr(this string s) {
			return new string(s.Reverse().ToArray());
		}

		public static int Gcd(int a, int b) {
			if (a < b)
				// 引数を入替えて自分を呼び出す
				return Gcd(b, a);
			while (b != 0) {
				var remainder = a % b;
				a = b;
				b = remainder;
			}
			return a;
		}
		public readonly static long MOD = 1000000007;

		public static string JoinString<T>(IEnumerable<T> array) {
			var sb = new StringBuilder();
			foreach (var item in array) {
				sb.Append(item);
				sb.Append(", ");
			}
			return sb.ToString();
		}

		public static void InitArray<T>(T[] ary, T value) {
			for (int i = 0; i < ary.Length; i++) {
				ary[i] = value;
			}
		}
		public static void InitDP2<T>(T[,] dp, T value) {
			for (int i = 0; i < dp.GetLength(0); i++) {
				for (int j = 0; j < dp.GetLength(1); j++) {
					dp[i, j] = value;
				}
			}
		}
		public static void InitDP3<T>(T[,,] dp, T value) {
			for (int i = 0; i < dp.GetLength(0); i++) {
				for (int j = 0; j < dp.GetLength(1); j++) {
					for (int k = 0; k < dp.GetLength(2); k++) {
						dp[i, j, k] = value;
					}
				}
			}
		}

		/// <summary>charでも対応可能なMax</summary>
		public static T Max<T>(T a, T b) where T : IComparable {
			return 0 <= a.CompareTo(b) ? a : b;
		}
		/// <summary>3要素以上に対応するMax</summary>
		public static T Max<T>(params T[] nums) where T : IComparable {
			if (nums.Length == 0) return default(T);

			T max = nums[0];
			for (int i = 1; i < nums.Length; i++) {
				max = max.CompareTo(nums[i]) > 0 ? max : nums[i];
			}
			return max;
		}
		/// <summary>charでも対応可能なMin</summary>
		public static T Min<T>(T a, T b) where T : IComparable {
			return 0 < a.CompareTo(b) ? b : a;
		}
		/// <summary>3要素以上に対応するMin</summary>
		public static T Min<T>(params T[] nums) where T : IComparable {
			if (nums.Length == 0) return default(T);

			T min = nums[0];
			for (int i = 1; i < nums.Length; i++) {
				min = min.CompareTo(nums[i]) < 0 ? min : nums[i];
			}
			return min;
		}

		///<summary>targetValueに一番近い値を返す</summary>
		public static long Nearest(long targetValue, params long[] values) {
			Debug.Assert(0 < values.Length);
			long minDiff = long.MaxValue;
			long ans = long.MaxValue;
			foreach (var v in values) {
				var diff = Math.Abs(v - targetValue);
				if (ReplaceIfSmaller(ref minDiff, diff)) {
					ans = v;
				}
			}
			return ans;
		}

		/// <summary>
		/// ソート済み配列 ary に同じ値の要素が含まれている？
		/// ※ソート順は昇順/降順どちらでもよい
		/// </summary>
		public static bool HasDuplicateInSortedArray<T>(T[] ary) where T : IComparable, IComparable<T> {
			if (ary.Length <= 1) return false;

			var lastNum = ary[ary.Length - 1];

			foreach (var n in ary) {
				if (lastNum.CompareTo(n) == 0) {
					return true;
				} else {
					lastNum = n;
				}
			}
			return false;
		}

		///<summary>v が r より大きい場合、r に v を代入し、trueを返す。それ以外（同値の場合を含む）は何もせずfalseを返す</summary>
		public static bool ReplaceIfBigger<T>(ref T r, T v) where T : IComparable {
			if (r.CompareTo(v) < 0) {
				r = v;
				return true;
			} else {
				return false;
			}
		}
		///<summary>v が r よりが小さい場合、r に v を代入し、trueを返す。それ以外（同値の場合を含む）は何もせずfalseを返す</summary>
		public static bool ReplaceIfSmaller<T>(ref T r, T v) where T : IComparable {
			if (0 < r.CompareTo(v)) {
				r = v;
				return true;
			} else {
				return false;
			}
		}

		public static void Swap<T>(ref T a, ref T b) where T : class {
			var tmp = a;
			a = b;
			b = tmp;
		}

		/// <summary>
		/// dic[key]にadderを加算する。keyが存在しなかった場合はdic[key]=adder をセットする。
		/// </summary>
		public static void AddTo<TKey>(this Dictionary<TKey, int> dic, TKey key, int adder) {
			if (dic.ContainsKey(key)) {
				dic[key] += adder;
			} else {
				dic[key] = adder;
			}
		}

		/// <summary>
		/// 文字列 s が chars に含まれる文字を含んでいるか？
		/// </summary>
		public static bool ContainsAny(this string s, char[] chars) {
			for (int j = 0; j < s.Length; j++) {
				if (chars.Contains(s[j])) return true;
			}
			return false;
		}

		/// <summary>
		/// 二分探索
		/// ※条件を満たす最小のidxを返す
		/// ※満たすものがない場合は ary.Length を返す
		/// ※『aryの先頭側が条件を満たさない、末尾側が条件を満たす』という前提
		/// ただし、IsOK(...)の戻り値を逆転させれば、逆でも同じことが可能。
		/// </summary>
		/// <param name="ary">探索対象配列 ★ソート済みであること</param>
		/// <param name="key">探索値 ※これ以上の値を持つ（IsOKがtrueを返す）最小のindexを返す</param>
		public static int BinarySearch<T>(T[] ary, T key) where T : IComparable, IComparable<T> {
			int left = -1;
			int right = ary.Length;

			while (1 < right - left) {
				var mid = left + (right - left) / 2;

				if (IsOK(ary, mid, key)) {
					right = mid;
				} else {
					left = mid;
				}
			}

			// left は条件を満たさない最大の値、right は条件を満たす最小の値になっている
			return right;
		}
		public static bool IsOK<T>(T[] ary, int idx, T key) where T : IComparable, IComparable<T> {
			// key <= ary[idx] と同じ意味
			return key.CompareTo(ary[idx]) <= 0;
		}

		/// <summary>
		/// nの 2進数における下からd(0-indexed)桁目のbitが立っている？
		/// </summary>
		public static int GetBit(long n, int d) {
			if (0 == (n & (1L << d)))
				return 0;
			else
				return 1;
		}
	}

	public class SolverBase
	{
		virtual protected string ReadLine() => Console.ReadLine();
		virtual protected string ReadString() => ReadLine();
		virtual protected int ReadInt() => int.Parse(ReadLine());
		virtual protected long ReadLong() => long.Parse(ReadLine());
		virtual protected string[] ReadStringArray() => ReadLine().Split(' ');
		virtual protected void ReadString2(out string a, out string b) {
			var ary = ReadStringArray();
			a = ary[0];
			b = ary[1];
		}
		virtual protected void ReadString3(out string a, out string b, out string c) {
			var ary = ReadStringArray();
			a = ary[0];
			b = ary[1];
			c = ary[2];
		}
		virtual protected char[] ReadCharArray() => ReadLine().Split(' ').Select<string, char>(s => s[0]).ToArray();
		virtual protected int[] ReadIntArray() => ReadLine().Split(' ').Select<string, int>(s => int.Parse(s)).ToArray();
		virtual protected void ReadInt2(out int a, out int b) {
			var ary = ReadIntArray();
			a = ary[0];
			b = ary[1];
		}
		virtual protected void ReadInt3(out int a, out int b, out int c) {
			var ary = ReadIntArray();
			a = ary[0];
			b = ary[1];
			c = ary[2];
		}
		virtual protected void ReadInt4(out int a, out int b, out int c, out int d) {
			var ary = ReadIntArray();
			a = ary[0];
			b = ary[1];
			c = ary[2];
			d = ary[3];
		}
		virtual protected long[] ReadLongArray() => ReadLine().Split(' ').Select<string, long>(s => long.Parse(s)).ToArray();
		virtual protected void ReadLong2(out long a, out long b) {
			var ary = ReadLongArray();
			a = ary[0];
			b = ary[1];
		}
		virtual protected void ReadLong3(out long a, out long b, out long c) {
			var ary = ReadLongArray();
			a = ary[0];
			b = ary[1];
			c = ary[2];
		}
		virtual protected void ReadLong4(out long a, out long b, out long c, out long d) {
			var ary = ReadLongArray();
			a = ary[0];
			b = ary[1];
			c = ary[2];
			d = ary[3];
		}
		virtual protected double[] ReadDoubleArray() => ReadLine().Split(' ').Select<string, double>(s => double.Parse(s)).ToArray();

		virtual protected void WriteLine(string line) => Console.WriteLine(line);
		virtual protected void WriteLine(double d) => Console.WriteLine($"{d:F9}");
		virtual protected void WriteLine<T>(T value) where T : IFormattable => Console.WriteLine(value);
		virtual protected void WriteGrid(IEnumerable<IEnumerable<char>> arrayOfArray) {
			var sb = new StringBuilder();
			foreach (var a in arrayOfArray) {
				foreach (var c in a) {
					sb.Append(c);
				}
				sb.AppendLine();
			}
			WriteLine(sb.ToString());
		}

		[Conditional("DEBUG")]
		protected void Dump(double d) => Console.WriteLine($"{d:F9}");
		[Conditional("DEBUG")]
		protected void Dump<T>(T x) => Console.WriteLine(x);
		[Conditional("DEBUG")]
		protected void DumpArray<T>(IEnumerable<T> array) {
			string s = Util.JoinString(array);
			// Consoleに出力すると、UnitTestの邪魔をしないというメリットあり。
			Console.WriteLine(s);
			//_writer.WriteLine(s);
		}
		[Conditional("DEBUG")]
		protected void DumpGrid(IEnumerable<IEnumerable<char>> arrayOfArray) {
			var sb = new StringBuilder();
			foreach (var a in arrayOfArray) {
				foreach (var c in a) {
					sb.Append(c);
				}
				sb.AppendLine();
			}
			// Consoleに出力すると、UnitTestの邪魔をしないというメリットあり。
			Console.WriteLine(sb.ToString());
			//_writer.WriteLine(sb.ToString());
		}
		[Conditional("DEBUG")]
		protected void DumpGrid<T>(IEnumerable<IEnumerable<T>> arrayOfArray) {
			var sb = new StringBuilder();
			foreach (var a in arrayOfArray) {
				sb.AppendLine(Util.JoinString(a));
			}
			// Consoleに出力すると、UnitTestの邪魔をしないというメリットあり。
			Console.WriteLine(sb.ToString());
			//_writer.WriteLine(sb.ToString());
		}
		[Conditional("DEBUG")]
		protected void DumpGrid(bool[,] grid) {
			var sb = new StringBuilder();
			for (int i = 0; i < grid.GetLength(0); i++) {
				for (int j = 0; j < grid.GetLength(1); j++) {
					sb.Append(grid[i, j] ? "x " : ". ");
				}
				sb.AppendLine();
			}
			Console.WriteLine(sb.ToString());
		}
		[Conditional("DEBUG")]
		protected void DumpGrid(char[,] grid) {
			var sb = new StringBuilder();
			for (int i = 0; i < grid.GetLength(0); i++) {
				for (int j = 0; j < grid.GetLength(1); j++) {
					sb.Append(grid[i, j]);
					sb.Append(" ");
				}
				sb.AppendLine();
			}
			Console.WriteLine(sb.ToString());
		}
		[Conditional("DEBUG")]
		protected void DumpDP<T>(T[,] dp) where T : IFormattable {
			var sb = new StringBuilder();
			for (int i = 0; i < dp.GetLength(0); i++) {
				for (int j = 0; j < dp.GetLength(1); j++) {
					sb.Append(dp[i, j]);
					sb.Append(", ");
				}
				sb.AppendLine();
			}
			Console.WriteLine(sb.ToString());
		}
		[Conditional("DEBUG")]
		protected void DumpDP3_Keta<T>(T[,,] dp) where T : IFormattable {
			var sb = new StringBuilder();
			for (int i = 0; i < dp.GetLength(0); i++) {
				sb.Append($"{i,2}: ");
				for (int j = 0; j < dp.GetLength(1); j++) {
					sb.Append($"{dp[i, j, 0]}-{dp[i, j, 1]}");
					sb.Append(", ");
				}
				sb.AppendLine();
			}
			Console.WriteLine(sb.ToString());
		}
	}
}