Number of ways to color boundary of each block of M*N table


Given a table of M * N. There are total M * N squares of size 1. You have to colour each side of all squares with 3 colour Orange, Blue or Black such that each square have 2 different colours and each colour must occur twice.
It means each square have four sides, 2 of them is orange and 2 of them is Blue or 2 of them is orange and 2 of them is Black or 2 of them is Blue and 2 of them is Black.

Examples:

Input: N = 1, M = 1
Output: 18
Explanation:
We can fill the upper part of the square and left part of the square with any of the three colours. So the number of ways is 3*3,
Now for filling the right and the lower part, we have only 2 option if upper and left have the same colour.
If the upper and the left part have different colour then we can use those two colours to fill the right and lower part, it has also 2 option. The number of combination to fill right and lower is 2.
A total possible way to colour the square is 3*3*2 = 18

Input: N = 3, M = 2
Output: 15552

Approach:



  • Find the number of ways to fill the upper and right side of the rectangle. Upper-side has M units and the right side has N units. So, The number of ways to colour the upper and the right side of the rectangle is pow(3, M + N) because each has 3 option.
  • Now, For filling the lower and the left side of each square, It has 2 option for each square then the number of ways is pow(2, M * N).
  • Final result is:

     Total count = pow(3, M + N ) *  pow(2, M * N)
    

Below is the implementation of the above approach:

C++

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// C++ program to count the number 
// of ways to color boundary of 
// each block of M*N table.
#include <bits/stdc++.h>
using namespace std;
  
// Function to compute all way to 
// fill the boundary of all sides
// of the unit square
int CountWays(int N, int M)
{
    int count = 1;
  
    // Count possible ways to fill
    // all upper and left side of 
    // the rectangle M*N
    count = pow(3, M + N);
  
    // Count possible ways to fill
    // all  side of the all squares 
    // unit size
    count *= pow(2, M * N);
  
    return count;
}
  
// Driver code
int main()
{
  
    // Number of rows
    int N = 3; 
      
    // Number of columns
    int M = 2; 
      
    cout << CountWays(N, M);
    return 0;
}

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Java

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// Java program to count the number 
// of ways to color boundary of 
// each block of M*N table.
import java.util.*; 
  
class GFG{ 
  
// Function to compute all way to 
// fill the boundary of all sides
// of the unit square
static int CountWays(int N, int M)
{
    int count = 1;
  
    // Count possible ways to fill
    // all upper and left side of 
    // the rectangle M*N
    count = (int)Math.pow(3, M + N);
  
    // Count possible ways to fill
    // all side of the all squares 
    // unit size
    count *= (int)Math.pow(2, M * N);
  
    return count;
}
  
// Driver code
public static void main(String args[])
{
  
    // Number of rows
    int N = 3
      
    // Number of columns
    int M = 2
      
    System.out.println(CountWays(N, M));
}
}
  
// This code is contributed by ANKITKUMAR34

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Python3

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# Python3 program to count the number 
# of ways to color boundary of 
# each block of M*N table.
  
# Function to compute all way to 
# fill the boundary of all sides
# of the unit square
def CountWays(N, M):
      
    count = 1
  
    # Count possible ways to fill
    # all upper and left side of 
    # the rectangle M*N
    count = pow(3, M + N)
  
    # Count possible ways to fill
    # all side of the all squares 
    # unit size
    count *= pow(2, M * N);
  
    return count
  
# Driver code
  
# Number of rows
N = 3
      
# Number of columns
M = 2
      
print(CountWays(N, M))
  
# This code is contributed by ANKITKUMAR34

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Output:

15552

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