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 Scala (2.13.1) (judge verdict: Accepted)

import java.security.KeyStore.TrustedCertificateEntry
import java.util
import java.util.regex.Pattern
import java.util.{ArrayList, Collections, Comparator, List => JList}

import scala.collection.immutable.TreeMap
import scala.math._
import scala.collection.mutable
import scala.collection.mutable.{ArrayBuffer, ListBuffer}
import scala.util.{Failure, Success, Try}

object Main{
  val LS = "\n"
  def main(args: Array[String]): Unit = {
    if (sys.env.getOrElse("TEST", "")=="1"){
      println(test());
    }else{
      val input=io.Source.stdin.getLines().mkString(LS);
      println(solve(input).trim());
    }
  }

  def init (nRows: Int, nCols: Int, default: Int) = Array.tabulate(nRows,nCols)( (x,y) =>  default)

  def sortIgnoreCase1(list: JList[String]): JList[String] = {
    val r = new ArrayList[String](list)
    Collections.sort(r, new Comparator[String] {
      override def compare(o1: String, o2: String): Int = o1.compareToIgnoreCase(o2)
    })
    r
  }

//  def solve(input: String): String = {
//    val Array(n, q) = input.split(LINE_SEPARATOR).head.split(" ").map(_.toInt)
//    val a = input.split(LINE_SEPARATOR).tail.take(n).map(_.split(" ").head.toInt)
//    val b = input.split(LINE_SEPARATOR).tail.take(n).map(_.split(" ").tail.head.toInt)
//    val c = input.split(LINE_SEPARATOR).drop(n + 1).map(_.split(" ").head.toInt)
//    val d = input.split(LINE_SEPARATOR).drop(n + 1).map(_.split(" ").tail.head.toInt)
//    val k = 2 * pow(10, 5).toInt
//    val INF = pow(10, 9).toInt + 1
//    val pqMap = (0 until k).map(_ => new mutable.TreeMap[Int, Int]).toArray
//    def add(k: Int, ai: Int): Unit = {
//      val count = pqMap(k).getOrElse(ai, 0)
//      pqMap(k) += ai -> (count + 1)
//    }
//    def delete(k: Int, ai: Int): Unit = {
//      val count = pqMap(k).getOrElse(ai, 0)
//      if (count == 1) pqMap(k) -= ai else pqMap(k) += ai -> (count - 1)
//    }
//    val i2k = (0 until n).map{i =>
//      b(i)
//    }.toArray
//    (0 until n).foreach{i =>
//      add(b(i) - 1, a(i))
//    }
//    val maxList = pqMap.map(pq => if (pq.nonEmpty) pq.last._1 else INF)
//
//    (0 until q).map{j =>
//      add(d(j) - 1, a(c(j) - 1))
//      delete(i2k(c(j) - 1) - 1, a(c(j) - 1))
//      maxList(i2k(c(j) - 1) - 1) =
//        if (pqMap(i2k(c(j) - 1) - 1).nonEmpty) pqMap(i2k(c(j) - 1) - 1).last._1 else INF
//      i2k(c(j) - 1) = d(j)
//      maxList(d(j) - 1) = if (pqMap(d(j) - 1).nonEmpty) pqMap(d(j) - 1).last._1 else INF
//      maxList.min
//    }.mkString(LINE_SEPARATOR)
//  }

  def solve(input: String): String = {
    val Array(h, w, k) = input.split(LS).head.split(" ").map(_.toInt)
    val c = input.split(LS).tail.map(_.toSeq.toArray).toArray

    val resList = (0 until 1 <<h).flatMap {bith =>
      (0 until 1 << w).map {bitw =>
        val res = c.map(_.clone)
        for (i <- (0 until h); j <- (0 until w)){
          if ((1 & (bith >> i)) == 1 || (1 & (bitw >> j)) == 1) {
            res(i)(j) = 'r'
          }
        }
        res.flatten.count(_ == '#')
      }
    }
    resList.count(_ == k).toString
  }

  def divideDigits(num: Long, base: Long = 10L): List[Long] = {
    def proc(num: Long, base: Long, flg: Boolean): List[Long] = {
      var flag = flg
      val quotient = num / base
      val remainder = if (num % base == 0) {
        flag = true
        base
      } else if (flag) {
        flag = false
        val temp = num % base - 1
        if (temp == 0) {
          flag = true
          base - 1
        } else temp
      } else {
        flag = flg
        num % base
      }
      quotient match {
        case 0 =>
          if (flag) {
            List.empty
          } else List(remainder)
        case _ => remainder :: proc(quotient, base, flag)
      }
    }

    proc(num, base, false).reverse//.filterNot(_ == 0)
  }


  def medianCalculator(seq: Seq[Long]): Long = {
    //In order if you are not sure that 'seq' is sorted
    val sortedSeq = seq.sortWith(_ < _)

    if (seq.size % 2 == 1) sortedSeq(sortedSeq.size / 2)
    else {
      val (up, down) = sortedSeq.splitAt(seq.size / 2)
      (up.last + down.head) / 2
    }
  }

  def factorize(x: Long): List[Long] = {
    def foo(x: Long, a: Long = 2, list: List[Long] = Nil): List[Long] = a*a > x match {
      case false if x % a == 0 => foo(x / a, a    , a :: list)
      case false               => foo(x    , a + 1, list)
      case true                => x :: list
    }
    foo(x)
  }

  def divisors(n: Int): Vector[Int] = {
    val halfList = (1 to sqrt(n).toInt).toVector.filter(n % _ == 0).map(_.toInt)
    val reversedList = if (sqrt(n).floor == sqrt(n)) halfList.reverse.tail.map(n/_) else halfList.reverse.map(n/_)
    halfList ++ reversedList
  }

  def gcd(m: Long, n: Long): Long = n match {
    case 0 => m
    case _ => gcd(n, m % n)
  }
  def lcm(a: Long, b: Long): Long = a * b / gcd(a, b)

  val tests=List(
    """2 3 2
      |..#
      |###""".stripMargin -> """5
                                   |""".stripMargin
    ,
    """2 3 4
      |..#
      |###""".stripMargin -> """1
                                |""".stripMargin,
    """2 2 3
      |##
      |##""".stripMargin -> """0""",
    """6 6 8
      |..##..
      |.#..#.
      |#....#
      |######
      |#....#
      |#....#""".stripMargin -> """208"""
  )

  def test():String= {
    return tests.map{case (i,o)=>(i.trim(),o.trim())}
      .zipWithIndex.map{
      case ((input,outputExpect),i)=>{
        val output=solve(input).trim();
        s"test${i+1}:"+(if(output==outputExpect){
          "Passed"
        }else{
          s"Failed\nexpect:\n${outputExpect}\noutput:\n${output}"
        })
      }}
      .mkString("\n");
  }



}