Count of non decreasing arrays of length N formed with values in range L to R

Given are integers N, L and R, the task is to count the number of non decreasing arrays of length N formed with values in range [L, R] with repetition allowed.

Examples:

Input: N = 4, L = 4, R = 6
Output: 5
All possible arrays are {4, 4, 4, 6}, {4, 4, 5, 6}, {4, 5, 5, 6}, {4, 5, 6, 6} and {4, 6, 6, 6}.

Input: N = 2, L = 5, R = 2
Output: 0
No such combinations exist as L > R.

Approach:



  • Since it is known that the minimum number is L and maximum number is R in the array.
  • If the remaining (N – 2) indices are filled with L, the minimum possible sum is obtained and if the remaining (N-2) indices are filled with R, the maximum possible sum is obtained.
  • It can be concluded that there exists a combination of numbers which results in a sum in between the minimum possible and maximum possible sum.
  • Therefore, total different number of sums can be computed by:
    [(N – 2) * R – (N – 2) * L] + 1 = (N – 2) * (R – L) + 1

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
  
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the count
// of different arrays
int countSum(int N, int L, int R)
{
  
    // No such combination exists
    if (L > R) {
        return 0;
    }
  
    // Arrays formed with single elements
    if (N == 1) {
        return R - L + 1;
    }
  
    if (N > 1) {
        return (N - 2) * (R - L) + 1;
    }
}
  
// Driver code
int main()
{
    int N = 4, L = 4, R = 6;
  
    cout << countSum(N, L, R);
  
    return 0;
}

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Java

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// Java implementation of the approach
class GFG
{
  
// Function to return the count
// of different arrays
static int countSum(int N, int L, int R)
{
  
    // No such combination exists
    if (L > R)
    {
        return 0;
    }
  
    // Arrays formed with single elements
    if (N == 1)
    {
        return R - L + 1;
    }
  
    if (N > 1)
    {
        return (N - 2) * (R - L) + 1;
    }
    return 0;
}
  
// Driver code
public static void main(String[] args)
{
    int N = 4, L = 4, R = 6;
  
    System.out.print(countSum(N, L, R));
}
}
  
// This code is contributed by 29AjayKumar

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Python3

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# Python3 implementation of the approach
  
# Function to return the count
# of different arrays
def countSum(N, L, R):
  
    # No such combination exists
    if (L > R):
        return 0;
  
    # Arrays formed with single elements
    if (N == 1):
        return R - L + 1;
    if (N > 1):
        return (N - 2) * (R - L) + 1;
      
    return 0;
  
# Driver code
if __name__ == '__main__':
    N, L, R = 4, 4, 6;
  
    print(countSum(N, L, R));
  
# This code is contributed by 29AjayKumar

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C#

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// C# implementation of the approach
using System;
  
class GFG
{
  
// Function to return the count
// of different arrays
static int countSum(int N, int L, int R)
{
  
    // No such combination exists
    if (L > R)
    {
        return 0;
    }
  
    // Arrays formed with single elements
    if (N == 1)
    {
        return R - L + 1;
    }
  
    if (N > 1)
    {
        return (N - 2) * (R - L) + 1;
    }
    return 0;
}
  
// Driver code
public static void Main(String[] args)
{
    int N = 4, L = 4, R = 6;
  
    Console.Write(countSum(N, L, R));
}
}
  
// This code is contributed by PrinciRaj1992

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Output:

5

Time Complexity: O(1)

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