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 Ruby (2.7.1) (judge verdict: Accepted)

# 0と1で表せる。
# hの選び方, 0,0, 1,0 0,1 1,1
# wの選び方, 0,0,0, 0,0,1, 0,1,0, 1,0,0, 0,1,1, 1,0,1, 1,1,0, 1,1,1

h,w,k=gets.split.map &:to_i
c = h.times.map { gets.chomp.chars }
c2 = Marshal.load(Marshal.dump(c))
res=0
=begin
#ph = [*0..h]
#pw = [*0..w]
pc = [h,w].max
p c
h.times do |i|
  w.times do |j|
   c[i][j]
  end
end
=end
ch = [0,1].repeated_permutation(h).to_a
cw = [0,1].repeated_permutation(w).to_a

# 組み合わせを全て試す。
ch.each do |i|
  cw.each do |j|
    # 赤く塗る作業
    # indexメソッドだと全ての1を取得できない
    #h_paint_idx = i.index(1)
    #h_paint_idx = j.index(1)
    i.each_with_index do |ii, eii|
      j.each_with_index do |jj, ejj|
        #if i == [0, 0] && j == [0, 1, 1]
        #  puts "=========#{jj}==and===#{j.index(jj)}===" 
        #end
        if ii == 1
          w.times do |wj|
            c[eii][wj] = 'red'
          end
        elsif  jj == 1
          h.times do |hi|
            c[hi][ejj] = 'red'
          end
        end # if end
      end # j
    end # i
    res += 1 if c.flatten.count('#') == k
    #p res
    #p c
    #puts "i:#{i}, j:#{j}"
    c = Marshal.load(Marshal.dump(c2))
    #p "##ターン##"
  end # cw
end # ch

puts res