C - H and V /

### 問題文

HW 列に並ぶマスからなるマス目があります。上から i 行目、左から j 列目 (1 \leq i \leq H, 1 \leq j \leq W) のマスの色は文字 c_{i,j} として与えられ、c_{i,j}. のとき白、# のとき黒です。

• 行を何行か選び (0 行でもよい)、列を何列か選ぶ (0 列でもよい)。そして、選んだ行に含まれるマスと、選んだ列に含まれるマスをすべて赤く塗る。

### 制約

• 1 \leq H, W \leq 6
• 1 \leq K \leq HW
• c_{i,j}. または #

### 入力

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}


### 入力例 1

2 3 2
..#
###


### 出力例 1

5


• 1 行目、1 列目
• 1 行目、2 列目
• 1 行目、3 列目
• 1 列目、2 列目
• 3 列目

### 入力例 2

2 3 4
..#
###


### 出力例 2

1


### 入力例 3

2 2 3
##
##


### 出力例 3

0


### 入力例 4

6 6 8
..##..
.#..#.
#....#
######
#....#
#....#


### 出力例 4

208


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