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Evaluate the expression ( N1 * (N – 1)2 * … * 1N) % (109 + 7)

  • Difficulty Level : Medium
  • Last Updated : 23 Apr, 2021

Given an integer N, the task is to find the value of the expression ( N1 * (N – 1)2 * … * 1N) % (109 + 7).

Input: N = 1 
Output:
Explanation: 
11 = 1

Input: N = 4 
Output: 288 
Explanation: 
41 * (4 – 1)2 * (4 – 2)3 * (4-3)4 
= 4 * 9 * 8 * 1 
= 288

Naive Approach: The simplest approach to solve this problem is to iterate over the range [1, N]. For every ith iteration, calculate the value of (N – i + 1)i. Finally, print the product of all the calculated values from each iteration. 
Time Complexity: O(N2 * log2(N)) 
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized based on the following observations:

F(N) = N1 * (N – 1)2 * … * 1N 
= N * (N – 1) * (N – 1) * (N – 2) * (N – 2) * (N – 2)* … 1 * 1 * 1 
= N * (N – 1) * (N – 2)*… * 1 * (N -1) * (N – 2) * …* 1 * … 
= N! * (N – 1)! * (N – 2)! * … * 1!



Follow the steps below to solve the problem:

  • Precompute the value of the factorial from 1 to N using factorial(N) = N * factorial(N – 1).
  • Iterate over the range [1, N] and find the product of all the factorials over the range [1, N] using the above observations
  • Finally, print the value of the expression.

C++




// C++ program to implement
// the above approach
 
#include <bits/stdc++.h>
using namespace std;
#define mod 1000000007
 
// Function to find the value of the expression
// ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
int ValOfTheExpression(int n)
{
 
    // factorial[i]: Stores factorial of i
    int factorial[n] = { 0 };
 
    // Base Case for factorial
    factorial[0] = factorial[1] = 1;
 
    // Precompute the factorial
    for (int i = 2; i <= n; i++) {
        factorial[i] = ((factorial[i - 1] % mod)
                        * (i % mod))
                       % mod;
    }
 
    // dp[N]: Stores the value of the expression
    // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
    int dp[n] = { 0 };
    dp[1] = 1;
 
    for (int i = 2; i <= n; i++) {
 
        // Update dp[i]
        dp[i] = ((dp[i - 1] % mod)
                 * (factorial[i] % mod))
                % mod;
    }
 
    // Return the answer.
    return dp[n];
}
 
// Driver Code
int main()
{
 
    int n = 4;
    // Function call
    cout << ValOfTheExpression(n) << "\n";
}

Java




// Java program to implement
// the above approach
class GFG
{
  static int mod = 1000000007;
 
  // Function to find the value of the expression
  // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
  static int ValOfTheExpression(int n)
  {
 
    // factorial[i]: Stores factorial of i
    int[] factorial = new int[n + 1];
 
    // Base Case for factorial
    factorial[0] = factorial[1] = 1;
 
    // Precompute the factorial
    for (int i = 2; i <= n; i++)
    {
      factorial[i] = ((factorial[i - 1] % mod)
                      * (i % mod)) % mod;
    }
 
    // dp[N]: Stores the value of the expression
    // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
    int[] dp = new int[n + 1];
    dp[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {
 
      // Update dp[i]
      dp[i] = ((dp[i - 1] % mod)
               * (factorial[i] % mod)) % mod;
    }
 
    // Return the answer.
    return dp[n];
  }
 
  // Driver code
  public static void main(String[] args) {
    int n = 4;
 
    // Function call
    System.out.println(ValOfTheExpression(n));
  }
}
 
// This code is contributed by divyesh072019

Python3




# Python 3 program to implement
# the above approach
mod = 1000000007
 
# Function to find the value of the expression
# ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
def ValOfTheExpression(n):
    global mod
     
    # factorial[i]: Stores factorial of i
    factorial = [0 for i in range(n + 1)]
 
    # Base Case for factorial
    factorial[0] = 1
    factorial[1] = 1
 
    # Precompute the factorial
    for i in range(2, n + 1, 1):
        factorial[i] = ((factorial[i - 1] % mod) * (i % mod))%mod
 
    # dp[N]: Stores the value of the expression
    # ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
    dp = [0 for i in range(n+1)]
    dp[1] = 1
    for i in range(2, n + 1, 1):
       
        # Update dp[i]
        dp[i] = ((dp[i - 1] % mod)*(factorial[i] % mod)) % mod
 
    # Return the answer.
    return dp[n]
 
# Driver Code
if __name__ == '__main__':
    n = 4
     
    # Function call
    print(ValOfTheExpression(n))
     
    # This code is contributed by SURENDRA_GANGWAR.

C#




// C# program to implement
// the above approach
using System;
class GFG
{
 
  static int mod = 1000000007;
 
  // Function to find the value of the expression
  // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
  static int ValOfTheExpression(int n)
  {
 
    // factorial[i]: Stores factorial of i
    int[] factorial = new int[n + 1];
 
    // Base Case for factorial
    factorial[0] = factorial[1] = 1;
 
    // Precompute the factorial
    for (int i = 2; i <= n; i++)
    {
      factorial[i] = ((factorial[i - 1] % mod)
                      * (i % mod))
        % mod;
    }
 
    // dp[N]: Stores the value of the expression
    // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
    int[] dp = new int[n + 1];
    dp[1] = 1;
 
    for (int i = 2; i <= n; i++)
    {
 
      // Update dp[i]
      dp[i] = ((dp[i - 1] % mod)
               * (factorial[i] % mod))
        % mod;
    }
 
    // Return the answer.
    return dp[n];
  }
 
  // Driver code
  static void Main()
  {
    int n = 4;
 
    // Function call
    Console.WriteLine(ValOfTheExpression(n));
  }
}
 
// This code is contributed by divyeshrabadiya07

Javascript




<script>
 
    // Javascript program to implement
    // the above approach
     
    let mod = 1000000007;
  
    // Function to find the value of the expression
    // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
    function ValOfTheExpression(n)
    {
 
      // factorial[i]: Stores factorial of i
      let factorial = new Array(n + 1);
 
      // Base Case for factorial
      factorial[0] = factorial[1] = 1;
 
      // Precompute the factorial
      for (let i = 2; i <= n; i++)
      {
        factorial[i] = ((factorial[i - 1] % mod) *
                        (i % mod)) % mod;
      }
 
      // dp[N]: Stores the value of the expression
      // ( N^1 * (N – 1)^2 * … * 1^N) % (109 + 7).
      let dp = new Array(n + 1);
      dp[1] = 1;
 
      for (let i = 2; i <= n; i++)
      {
 
        // Update dp[i]
        dp[i] = ((dp[i - 1] % mod) *
                 (factorial[i] % mod)) % mod;
      }
 
      // Return the answer.
      return dp[n];
    }
     
    let n = 4;
  
    // Function call
    document.write(ValOfTheExpression(n));
   
</script>
Output: 
288

 

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

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