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++ (GCC 9.2.1) (judge verdict: Accepted)

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned long long ull;
#define fo(i,n) for(ll i=0;i<n;i++)
#define fok(i,k,n) for(ll i=k;i<n;i++)
#define ci(x) cin>>x
#define ci2(x,y) cin>>x>>y
#define co(x) cout<<x<<"\n"
#define co2(x,y) cout<<x<<" "<<y<<"\n"
#define pb push_back
#define pf push_front
#define mp make_pair
#define fi first
#define se second
#define PI 3.1415926535897932384626
#define deb1(x) cout<<#x<<" "<<x<<endl
#define deb2(x,y) cout<<#x<<" "<<x<<" "<<#y<<" "<<y<<"\n"
#define deb3(x,y,z) cout<<#x<<" "<<x<<" "<<#y<<" "<<y<<" "<<#z<<" "<<z<<"\n"
typedef pair<int, int>	pii;
typedef pair<ll, ll>	pll;
typedef vector<int>		vi;
typedef vector<ll>		vl;
typedef vector<pii>		vpii;
typedef vector<pll>		vpll;
typedef vector<vi>		vvi;
typedef vector<vl>		vvl;
const int mod = 1000000007;	
int gcd(int a, int b){ if(a==0) return b; return gcd(b%a, a);}

int main(){
	ios_base::sync_with_stdio(false);
	cin.tie(NULL);
	int h ,w,k;
	cin>>h>>w>>k;
	int arr[h][w];
	fo(i,h){
		fo(j,w){
			char tmp;
			ci(tmp);
			if(tmp=='.') arr[i][j]=0;
			else arr[i][j]=1;
		}
	}
	int cnt=0;
	fo(y,(1<<h)){
		fo(x,(1<<w)){
			int black=0;
			fo(i,h){
				if(((y>>i)&1)==0) continue;
				fo(j,w){
					if(((x>>j)&1)==0) continue;
					black+=arr[i][j];
				}
			}
			if(black==k) cnt++;
		}
	}
	co(cnt);
	return 0;
}