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 Java (OpenJDK 11.0.6) (judge verdict: Accepted)

import java.io.*;
import java.util.*;
public class Main {
 
  public static void main(String[] args) { 
        solve();
      out.close();
  }

  private static void solve() {
      int h = sc.nextInt();
      int w = sc.nextInt();
      int k = sc.nextInt();
      int res = 0;
      char[][] grid = new char[h][w];
      for(int i = 0; i < h; i++) {
        grid[i] = sc.next().toCharArray();
      }
      for(int i = 0; i < (1 << h); i++) {
        for(int j = 0; j < (1 << w); j++) {
          int cnt = 0;
          for(int row = 0; row < h; row++) {
            for(int col = 0; col < w; col++) {
              if(((i >> row) & 1) == 1 || ((j >> col) & 1) == 1) continue;
              if(grid[row][col] == '#') cnt++;
            }
          }
          if(cnt == k)
            res++;
        }
      }
      out.println(res);
  }

  static int mod = 1_000_000_007;

  public static PrintWriter out = new PrintWriter(new BufferedOutputStream(System.out));
 
  public static FastScanner sc = new FastScanner();

  static class FastScanner {
      BufferedReader br=new BufferedReader(new InputStreamReader(System.in));
      StringTokenizer st=new StringTokenizer("");
      String next() {
        while (!st.hasMoreTokens())
          try {
            st=new StringTokenizer(br.readLine());
          } catch (IOException e) {
            e.printStackTrace();
          }
        return st.nextToken();
      }
      
      int nextInt() {
        return Integer.parseInt(next());
      }
      int[] readArray(int n) {
        int[] a=new int[n];
        for (int i=0; i<n; i++) a[i]=nextInt();
        return a;
      }
      long nextLong() {
        return Long.parseLong(next());
      }
  }

  static long pow(long a, long N) {
    if (N == 0) return 1;
    else if (N == 1) return a;
    else {
      long R = pow(a,N/2);
      if (N % 2 == 0) {
        return R*R;
      }
      else {
        return R*R*a;
      }
    }
  }
   
  static long powMod(long a, long N) {
    if (N == 0) return 1;
    else if (N == 1) return a % mod;
    else {
      long R = powMod(a,N/2) % mod;
      R *= R % mod;
      if (N % 2 == 1) {
        R *= a % mod;
      }
        return R % mod;
    }
  }

  static void mergeSort(int[] A){ // low to hi sort, single array only
    int n = A.length;
    if (n < 2) return;
    int[] l = new int[n/2];
    int[] r = new int[n - n/2];

    for (int i = 0; i < n/2; i++){
      l[i] = A[i];
    }

    for (int j = n/2; j < n; j++){
      r[j-n/2] = A[j];
    }

    mergeSort(l);
    mergeSort(r);
    merge(l, r, A);
  }
 
  static void merge(int[] l, int[] r, int[] a){
    int i = 0, j = 0, k = 0;
    while (i < l.length && j < r.length && k < a.length){
      if (l[i] < r[j]){
        a[k] = l[i];
        i++;
      }
      else{
        a[k] = r[j];
        j++;
      }
      k++;
    }
    while (i < l.length){
      a[k] = l[i];
      i++;
      k++;
    }

    while (j < r.length){
      a[k] = r[j];
      j++;
      k++;
    }
  }

  private static class Edge {
    int src;
    int dest;
    int weight;
    public Edge(int s, int d, int w) {
      this.src = s;
      this.dest = d;
      this.weight = w;
    }
  }
  private static int[] prim(Map<Integer, List<Edge>> graph) {
      int n = graph.size(); // number of the nodes
      Edge[] edges = new Edge[n];
      for(int i = 0; i < n; i++)
        edges[i] = new Edge(-1, i, Integer.MAX_VALUE);
      edges[0].weight = 0;

      int[] parents = new int[n];
      parents[0] = -1;

      TreeSet<Edge> treeSet = new TreeSet<>((a, b) -> a.weight - b.weight);
      treeSet.add(edges[0]);

      boolean[] visited = new boolean[n];
      while(treeSet.size() > 0) {
        Edge e = treeSet.pollFirst();
        visited[e.dest] = true;
        for(Edge child : graph.get(e.dest)) {
          if(visited[child.dest]) continue;
          if(edges[child.dest].weight > child.weight) {
            treeSet.remove(edges[child.dest]);
            edges[child.dest].weight = child.weight;
            treeSet.add(edges[child.dest]);
            parents[child.dest] = e.dest;
          }
        }
      }
      return parents;
  }

  private static Edge[] kruskal(int v, Edge[] edges) {
    Edge[] res = new Edge[v-1];

    int[] p = new int[v];
    for(int i = 0; i < v; i++) 
      p[i] = i;

    int[] rank = new int[v];
    Arrays.fill(rank, 1);

    Arrays.sort(edges, (e1, e2) -> e1.weight - e2.weight);
    for(int i = 0, idx = 0; i < res.length;) {
      Edge e = edges[idx++];
      if(union(p, rank, e.src, e.dest)) {
        res[i++] = e;
      }
    }
    return res;
  }

  private static boolean union(int[] root, int[] rank, int u, int v) {
      int rootA = find(root, u);
      int rootB = find(root, v);
      if(rootA == rootB) return false;

      if(rank[rootA] > rank[rootB]) {
        rank[rootA] += rank[rootB];
        root[rootB] = rootA;
      } else {
        rank[rootB] += rank[rootA];
        root[rootA] = rootB;
      }
      return true;
  }

  private static int find(int[] root, int node) {
    if(root[node] != node) {
      root[node] = find(root, root[node]);
    }
    return root[node];
  }
}