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Count N-length arrays made from first M natural numbers whose subarrays can be made palindromic by replacing less than half of its elements

  • Difficulty Level : Medium
  • Last Updated : 16 Apr, 2021

Given two integer N and M, the task is to find the count of arrays of size N with elements from the range [1, M] in which all subarrays of length greater than 1 can be made palindromic by replacing less than half of its elements i.e., floor(length/2).

Examples: 

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Input: N = 2, M = 3
Output: 6
Explanation:
There are 9 arrays possible of length 2 using values 1 to 3 i.e. [1, 1], [1, 2], [1, 3], [2, 1][2, 2], [2, 3], [3, 1], [3, 2], [3, 3].
All of these arrays except [1, 1], [2, 2] and [3, 3] have subarrays of length greater than 1 which requires 1 operation to make them palindrome. So the required answer is 9 – 3 = 6.



Input: N = 5, M = 10
Output: 30240

Approach: The problem can be solved based on the following observations:

  • It is possible that the maximum permissible number of operations required to make an array a palindrome is floor(size(array)/2).
  • It can be observed that by choosing a subarray, starting and ending with the same value, the number of operations needed to make it a palindrome will be less than floor(size of subarray)/2.
  • Therefore, the task is reduced to finding the number of arrays of size N using integer values in the range [1, M], which do not contain any duplicate elements, which can be easily done by finding the permutation of M with N i.e. MpN, which is equal to M * (M – 1) * (M – 2) * … * (M – N + 1).

Follow the steps below to solve the problem:

  1. Initialize an integer variable, say ans = 1.
  2. Traverse from i = 0 to N – 1 and update ans as ans = ans * (M-i)
  3. Print ans as the answer.

Below is the implementation of the above approach:

C++




// C++ program for the above approach
 
#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
 
// Function to find the number of arrays
// following the given condition
void noOfArraysPossible(ll N, ll M)
{
    // Initialize answer
    ll ans = 1;
 
    // Calculate nPm
    for (ll i = 0; i < N; ++i) {
        ans = ans * (M - i);
    }
 
    // Print ans
    cout << ans;
}
 
// Driver Code
int main()
{
 
    // Given N and M
    ll N = 2, M = 3;
 
    // Function Call
    noOfArraysPossible(N, M);
 
    return 0;
}

Java




// Java program for the above approach
class GFG
{
 
// Function to find the number of arrays
// following the given condition
static void noOfArraysPossible(int N, int M)
{
    // Initialize answer
    int ans = 1;
 
    // Calculate nPm
    for (int i = 0; i < N; ++i)
    {
        ans = ans * (M - i);
    }
 
    // Print ans
    System.out.print(ans);
}
 
// Driver Code
public static void main(String[] args)
{
 
    // Given N and M
    int N = 2, M = 3;
 
    // Function Call
    noOfArraysPossible(N, M);
}
}
 
// This code is contributed by Princi Singh

Python3




# Python3 program for the above approach
 
# Function to find the number of arrays
# following the given condition
def noOfArraysPossible(N, M):
     
    # Initialize answer
    ans = 1
  
    # Calculate nPm
    for i in range(N):
        ans = ans * (M - i)
         
    # Print ans
    print(ans)
  
# Driver Code
if __name__ == "__main__" :
     
    # Given N and M
    N = 2
    M = 3
     
    # Function Call
    noOfArraysPossible(N, M)
     
# This code is contributed by jana_sayantan

C#




// C# program to implement
// the above approach 
using System;
 
class GFG{
      
// Function to find the number of arrays
// following the given condition
static void noOfArraysPossible(int N, int M)
{
    // Initialize answer
    int ans = 1;
  
    // Calculate nPm
    for (int i = 0; i < N; ++i)
    {
        ans = ans * (M - i);
    }
  
    // Print ans
    Console.Write(ans);
}
  
// Driver Code
public static void Main()
{
    // Given N and M
    int N = 2, M = 3;
  
    // Function Call
    noOfArraysPossible(N, M);
}
}
 
// This code is contributed by susmitakundugoaldanga

Javascript




<script>
// javascript program for the above approach   
// Function to find the number of arrays
 
    // following the given condition
    function noOfArraysPossible(N , M)
    {
     
        // Initialize answer
        var ans = 1;
 
        // Calculate nPm
        for (i = 0; i < N; ++i) {
            ans = ans * (M - i);
        }
 
        // Prvar ans
        document.write(ans);
    }
 
    // Driver Code
     
        // Given N and M
        var N = 2, M = 3;
 
        // Function Call
        noOfArraysPossible(N, M);
 
// This code is contributed by todaysgaurav
</script>
Output: 
6

 

Time Complexity: O(N) 
Auxiliary Space: O(1)

 




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