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 Haskell (GHC 8.8.3) (judge verdict: Accepted)

{-# LANGUAGE
    BangPatterns, MultiWayIf, TupleSections #-}
{-# OPTIONS_GHC -O2 #-}


import Control.Arrow
import Control.Concurrent
import Control.Monad
import Data.Bits
import qualified Data.ByteString.Builder as B
import qualified Data.ByteString.Char8 as C
import Data.Char
import Data.Complex
import Data.Function
import Data.Function.HT
import qualified Data.IntMap.Strict as IM
import qualified Data.IntSet as IS
import Data.List
import Data.Maybe
import Data.Monoid
import Data.Ord
import qualified Data.Sequence as Sq
import Data.Vector.Algorithms.Search
import qualified Data.Vector as V
import qualified Data.Vector.Mutable as VM
import qualified Data.Vector.Unboxed as U
import qualified Data.Vector.Unboxed.Mutable as UM
import Debug.Trace
import System.IO


r :: Int -> IO (U.Vector Int)
r n = U.unfoldrN n (C.readInt . C.dropWhile (==' ')) <$> C.getLine

r' :: IO [Int]
r'  =   unfoldr    (C.readInt . C.dropWhile (==' ')) <$> C.getLine


main :: IO ()
main = do
    [h,w,k] <- r'
    cs <- (V.fromList . concat) <$> replicateM h getLine
    let choices = forM (V.replicate (h + w) (0 :: Int)) (\_ -> [1,0])
    print $ length $ filter (solve cs (h,w,k)) choices

solve :: V.Vector Char -> (Int, Int, Int) -> V.Vector Int -> Bool
solve board (h,w,k) choice =
    k == (V.sum $ V.imap (\i c -> let (h',w') = quotRem i w
        in if c == '#' && (choice V.! h') == 0 && choice V.! (h + w') == 0 then 1 else 0) board)