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.Collections.Immutable;
using System.Diagnostics;
using System.IO;
using System.Linq;
using System.Numerics;
using System.Runtime.InteropServices;
using System.Text;
using System.Text.RegularExpressions;
using System.Threading.Tasks;
using static System.Math;
using static Math2;
using static Output;
using static Utils;

public class CP
{
    Input sc;
    public void Solve()
    {
        sc = new Input();
        bool multicases = false;
        var t = multicases ? sc.Int : 1;
        // var sw = new Stopwatch();
        // sw.Start();
        while (t-- > 0)
        {
            case1();
        }
        // sw.Stop();
        //Put(sw.ElapsedMilliseconds + "ms");
    }
    public void case1()
    {
        int h = sc.Int;
        int w = sc.Int;
        int k = sc.Int;
        var c = Enumerable.Range(0, h).Select(_ => sc.Chararr).ToArray();
        int ans = 0;

        for (int i = 0; i < (1 << (h + w)); ++i)
        {
            var nh = new List<int>();
            for (int j = 0; j < h; ++j)
            {
                if (((i >> j) & 1) == 1) nh.Add(j);
            }
            var nw = new List<int>();
            for (int j = 0; j < w; ++j)
            {
                var jj = j + h;
                if (((i >> jj) & 1) == 1) nw.Add(j);
            }
            var banmen = new char[h][];
            for (int j = 0; j < h; ++j) banmen[j] = c[j].ToArray();
            foreach (var x in nh)
            {
                for (int l = 0; l < w; ++l)
                {
                    banmen[x][l] = 'q';
                }
            }
            foreach (var x in nw)
            {
                for (int l = 0; l < h; ++l)
                {
                    banmen[l][x] = 'q';
                }
            }
            int cnt = 0;
            for (int j = 0; j < h; ++j)
            {
                cnt += banmen[j].Count(z => z == '#');
            }
            if (cnt == k) ++ans;
        }
        Put(ans);
    }
    public void case2()
    {
        int n = sc.Int;
        var a = sc.Intarr;
        Array.Sort(a, (y, x) => x - y);
        long ans = 0;
        int l = 0;
        int r = 0;
        for (int i = 0; i < n; ++i)
        {
            ans += Min(l, r);
            if (i == 0)
            {
                l = a[i];
                r = a[i];
            }

        }
    }
}
public class PriorityQueue<T>
{
    List<T> _item;
    public int Count { get { return _item.Count; } }
    bool _isascend { get; set; }
    public T Peek { get { return _item[0]; } }
    Comparison<T> Comp;
    public PriorityQueue(bool IsAscend = true, IEnumerable<T> list = null) : this(Comparer<T>.Default.Compare, IsAscend, list) { }
    public PriorityQueue(Comparison<T> cmp, bool IsAscend = true, IEnumerable<T> list = null)
    {
        _item = new List<T>();
        _isascend = IsAscend;
        this.Comp = cmp;
        if (list != null)
        {
            _item.AddRange(list);
            Build();
        }
    }

    private int Compare(int i, int j) => (_isascend ? -1 : 1) * Comp(_item[i], _item[j]);
    private void Swap(int i, int j) { var t = _item[i]; _item[i] = _item[j]; _item[j] = t; }
    private int Parent(int i) => (i - 1) >> 1;
    private int Left(int i) => (i << 1) + 1;
    public T Enqueue(T val)
    {
        int i = _item.Count;
        _item.Add(val);
        while (i > 0)
        {
            int p = Parent(i);
            if (Compare(i, p) > 0)
                Swap(i, p);
            i = p;
        }
        return val;
    }
    private void Heapify(int index)
    {
        for (int i = index, j;
            (j = Left(i)) < _item.Count; i = j)
        {
            if (j != _item.Count - 1 && Compare(j, j + 1) < 0) j++;
            if (Compare(i, j) < 0)
                Swap(i, j);
        }
    }
    public T Dequeue()
    {
        T val = _item[0];
        _item[0] = _item[_item.Count - 1];
        _item.RemoveAt(_item.Count - 1);
        Heapify(0);
        return val;
    }
    private void Build()
    {
        for (var i = (_item.Count >> 1) - 1; i >= 0; i--)
            Heapify(i);
    }
    public bool Any() => Count > 0;
}
public class BinarySearch
{
    public static int Lower_Bound<T>(IList<T> Array, T Target)
    {
        return Lower_Bound<T>(Array, Target, Comparer<T>.Default.Compare);
    }
    public static int Lower_Bound<T>(IList<T> Array, T Target, Comparison<T> Comp)
    {
        var cmp = Comparer<T>.Create(Comp);
        var l = -1; //always ng
        var r = Array.Count(); //always ok
        while (r - l > 1)
        {
            var mid = l + (r - l) / 2;
            var res = cmp.Compare(Array[mid], Target);
            if (res >= 0) r = mid;
            else l = mid;
        }
        return r;
    }
    public static int Upper_Bound<T>(IList<T> Array, T Target)
    {
        return Upper_Bound<T>(Array, Target, Comparer<T>.Default.Compare);
    }
    public static int Upper_Bound<T>(IList<T> Array, T Target, Comparison<T> Comp)
    {
        var cmp = Comparer<T>.Create(Comp);
        var l = -1;
        var r = Array.Count;
        while (r - l > 1)
        {
            var mid = l + (r - l) / 2;
            var res = cmp.Compare(Array[mid], Target);
            if (res > 0) r = mid;
            else l = mid;
        }
        return r;
    }
    public static bool Contains<T>(IList<T> Arr, T Target)
    {
        return Contains<T>(Arr, Target, Comparer<T>.Default.Compare);
    }
    public static bool Contains<T>(IList<T> Arr, T Target, Comparison<T> Comp)
    {
        return (Upper_Bound(Arr, Target, Comp) - Lower_Bound(Arr, Target, Comp) > 0);
    }
}
public static class Math2
{
    public const int INF = 1 << 29;
    public const long INFL = 1L << 60;
    public const long MOD = 1000000007;
    public const long MOD2 = 998244353;

    public static long Power(long a, long b, long MOD = 1000000007) //i^N
    {
        long res = 1;
        while (b > 0)
        {
            if ((b & 1) != 0) res = res * a % MOD;
            a = a * a % MOD;
            b >>= 1;
        }
        return res;
    }
    public static long Power2(long a, long b) //i^N
    {
        long res = 1;
        while (b > 0)
        {
            if ((b & 1) != 0) res = res * a;
            a = a * a;
            b >>= 1;
        }
        return res;
    }
    public static long GCD(long a, long b)
    {
        if (a == 0) return b;
        while (b > 0)
        {
            var r = a % b;
            a = b;
            b = r;
        }
        return a;
    }
    public static long LCM(long a, long b) => (b / GCD(a, b)) * a;
    public static long Comb(long n, long r, int MOD = 1000000007)
    {
        if (r > n - r) r = n - r;
        long Nume = 1;
        long Deno = 1;

        if (r > n - r) r = n - r;
        for (long i = 1; i <= r; ++i)
        {
            Deno = (Deno * i) % MOD;
            Nume = Nume * (n - i + 1) % MOD;
        }
        return (Nume * inv(Deno)) % MOD;
    }
    public static long Comb2(long n, long r)
    {
        long Nume = 1;
        long Deno = 1;

        if (r > n - r) r = n - r;
        for (long i = 1; i <= r; ++i)
        {
            Deno *= i;
            Nume *= n - i + 1;
        }
        return Nume / Deno;
    }
    public static long inv(long x, int MOD = 1000000007)
    {
        return Power(x, MOD - 2, MOD);
    }
}
public static class Utils
{

    public static T[] sort<T>(this T[] a)
    {
        Array.Sort(a);
        return a;
    }
    public static int popcnt(long a)
    {
        int cnt = 0;
        while (a > 0)
        {
            if ((a & 1) == 1) ++cnt;
            a >>= 1;
        }
        return cnt;
    }
    public static void Swap<T>(ref T A, ref T B)
    {
        T x = A;
        A = B;
        B = x;
    }
    public static int DigitSum(string N)
    {

        int ret = 0;
        for (int i = 0; i < N.Length; ++i) ret += N[i] - '0';
        return ret;
    }
    public static string ConvertBase(long N, int K)
    {
        StringBuilder ret = new StringBuilder();
        while (N > 0)
        {
            var r = N % K;
            N /= K;
            ret.Append(r);
        }
        return new string(ret.ToString().ToCharArray().Reverse().ToArray());
    }
    public static bool NextPermutation<T>(IList<T> lis, Comparison<T> cmp)
    {
        int n = lis.Count;
        int i = n - 1;
        while (i - 1 >= 0)
        {
            if (cmp(lis[i - 1], lis[i]) < 0) break;
            --i;
        }
        if (i == 0) return false;
        int j = i;
        while (j + 1 < n)
        {
            if (cmp(lis[i - 1], lis[j + 1]) > 0) break;
            ++j;
        }
        var _q = lis[j];
        lis[j] = lis[i - 1];
        lis[i - 1] = _q;
        int k = i;
        int l = n - 1;
        while (k < l)
        {
            var _p = lis[k];
            lis[k] = lis[l];
            lis[l] = _p;

            ++k;
            --l;
        }
        return true;
    }
    public static bool NextPermutation<T>(IList<T> lis) => NextPermutation(lis, Comparer<T>.Default.Compare);
}
class Program
{
    static void Main(string[] args)
    {
        var CP = new CP();
        CP.Solve();
    }
}
public static class Output
{

    public static void Put(string a) => Console.WriteLine(a);
    public static void Put(params object[] i) => Put(string.Join(" ", i));
    public static void Put<T>(IEnumerable<T> a) => Put(string.Join(" ", a));
    public static void PutV<T>(IEnumerable<T> a) { foreach (var z in a) Put(z); }
    public static void YN(bool i)
    {
        if (i) Put("Yes");
        else Put("No");
    }
}
public class Input
{
    public static string Str => Console.ReadLine();
    public static bool IsTypeEqual<T, U>() => typeof(T).Equals(typeof(U));
    public static T ConvertType<T, U>(U a) => (T)Convert.ChangeType(a, typeof(T));
    public static T Cast<T>(string s)
    {
        if (IsTypeEqual<T, int>()) return ConvertType<T, int>(int.Parse(s));
        else if (IsTypeEqual<T, long>()) return ConvertType<T, long>(long.Parse(s));
        else if (IsTypeEqual<T, double>()) return ConvertType<T, double>(double.Parse(s));
        else if (IsTypeEqual<T, char>()) return ConvertType<T, char>(char.Parse(s));
        else if (IsTypeEqual<T, BigInteger>()) return ConvertType<T, BigInteger>(BigInteger.Parse(s));
        else if (IsTypeEqual<T, decimal>()) return ConvertType<T, decimal>(decimal.Parse(s));
        else return ConvertType<T, string>(s);
    }
    public static T[] Castarr<T>(string[] s)
    {

        var ret = new T[s.Length];
        int i = 0;
        if (IsTypeEqual<T, char>())
        {
            var list = new List<T>();
            foreach (var t in s)
            {
                foreach (var u in t)
                {
                    list.Add(ConvertType<T, char>(char.Parse(u.ToString())));
                }
            }
            return list.ToArray();
        }
        foreach (var t in s)
        {
            if (IsTypeEqual<T, int>()) ret[i++] = ConvertType<T, int>(int.Parse(t));
            else if (IsTypeEqual<T, long>()) ret[i++] = ConvertType<T, long>(long.Parse(t));
            else if (IsTypeEqual<T, double>()) ret[i++] = ConvertType<T, double>(double.Parse(t));
            else if (IsTypeEqual<T, BigInteger>()) ret[i++] = ConvertType<T, BigInteger>(BigInteger.Parse(t));
            else ret[i++] = ConvertType<T, string>(t);
        }
        return ret;
    }
    Queue<string> q = new Queue<string>();
    void next() { var ss = Str.Split(' '); foreach (var item in ss) q.Enqueue(item); }
    public T cin<T>() { if (!q.Any()) next(); return Cast<T>(q.Dequeue()); }
    public T[] cinarr<T>() { return Castarr<T>(Str.Split(' ')); }
    public T[] cinarr<T>(int n) { var ret = new T[n]; for (int i = 0; i < n; ++i) ret[i] = cin<T>(); return ret; }
    public int Int => cin<int>();
    public long Long => cin<long>();
    public double Double => cin<double>();
    public char Char => cin<char>();
    public string String => cin<string>();
    public BigInteger BI => cin<BigInteger>();
    public int[] Intarr => cinarr<int>();
    public long[] Longarr => cinarr<long>();
    public double[] Doublearr => cinarr<double>();
    public char[] Chararr => cinarr<char>();
    public string[] Stringarr => cinarr<string>();
    public BigInteger[] BIarr => cinarr<BigInteger>();
    public void cin<T>(out T t) { t = cin<T>(); }
    public void mul<T, U>(out T t, out U u) { t = cin<T>(); u = cin<U>(); }
    public void mul<T, U, V>(out T t, out U u, out V v) { t = cin<T>(); u = cin<U>(); v = cin<V>(); }
    public void mul<T, U, V, W>(out T t, out U u, out V v, out W w) { t = cin<T>(); u = cin<U>(); v = cin<V>(); w = cin<W>(); }
    public void mul<T, U, V, W, X>(out T t, out U u, out V v, out W w, out X x) { t = cin<T>(); u = cin<U>(); v = cin<V>(); w = cin<W>(); x = cin<X>(); }
    public void mul<T, U, V, W, X, Y>(out T t, out U u, out V v, out W w, out X x, out Y y) { t = cin<T>(); u = cin<U>(); v = cin<V>(); w = cin<W>(); x = cin<X>(); y = cin<Y>(); }
    public void mul<T, U, V, W, X, Y, Z>(out T t, out U u, out V v, out W w, out X x, out Y y, out Z z) { t = cin<T>(); u = cin<U>(); v = cin<V>(); w = cin<W>(); x = cin<X>(); y = cin<Y>(); z = cin<Z>(); }
}