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Product of first N factorials
  • Last Updated : 23 Aug, 2019

Given a number N. Find the product of first N factorials modulo 1000000007.

Constraints: 1 ≤ N ≤ 1e6

Examples:

Input : 3
Output : 12
Explanation: 1! * 2! * 3! = 12 mod (1e9 + 7) = 12

Input : 5
Output : 34560

Prerequisites: Modular Multiplication

Approach: The basic idea behind solving this problem is to just consider the problem of overflow during multiplication of such large numbers i.e. factorials. Hence, it needs to be addressed by multiplying recursively to overcome the difficulty of overflow. Moreover, we have to take modulus at every step while computing factorials iteratively and modular multiplication.



facti = facti-1 * i
where facti is the factorial of ith number

prodi = prodi-1 * facti
where prodi is the product of first i factorials

To find product of two large numbers under modulo, we use same approach as exponentiation under modulo.. In the multiplication function, we use + instead of *.

Below is the implementation of above approach.

C++




// CPP Program to find the
// product of first N factorials
#include <bits/stdc++.h>
  
using namespace std;
  
// To compute (a * b) % MOD
long long int mulmod(long long int a, long long int b, 
                                    long long int mod)
{
    long long int res = 0; // Initialize result
    a = a % mod;
    while (b > 0) {
  
        // If b is odd, add 'a' to result
        if (b % 2 == 1)
            res = (res + a) % mod;
  
        // Multiply 'a' with 2
        a = (a * 2) % mod;
  
        // Divide b by 2
        b /= 2;
    }
  
    // Return result
    return res % mod;
}
  
// This function computes factorials and
// product by using above function i.e.
// modular multiplication
long long int findProduct(long long int N)
{
    // Initialize product and fact with 1
    long long int product = 1, fact = 1;
    long long int MOD = 1e9 + 7;
    for (int i = 1; i <= N; i++) {
  
        // ith factorial
        fact = mulmod(fact, i, MOD);
  
        // product of first i factorials
        product = mulmod(product, fact, MOD);
  
        // If at any iteration, product becomes
        // divisible by MOD, simply return 0;
        if (product == 0)
            return 0;
    }
    return product;
}
  
// Driver Code to Test above functions
int main()
{
    long long int N = 3;
    cout << findProduct(N) << endl;
  
    N = 5;
    cout << findProduct(N) << endl;
  
    return 0;
}

Java




// Java Program to find the
// product of first N factorials
  
class GFG{
// To compute (a * b) % MOD
static double mulmod(long a, long b, 
                                    long mod)
{
    long res = 0; // Initialize result
    a = a % mod;
    while (b > 0) {
  
        // If b is odd, add 'a' to result
        if (b % 2 == 1)
            res = (res + a) % mod;
  
        // Multiply 'a' with 2
        a = (a * 2) % mod;
  
        // Divide b by 2
        b /= 2;
    }
  
    // Return result
    return res % mod;
}
  
// This function computes factorials and
// product by using above function i.e.
// modular multiplication
static long findProduct(long N)
{
    // Initialize product and fact with 1
    long product = 1, fact = 1;
    long MOD = (long)(1e9 + 7);
    for (int i = 1; i <= N; i++) {
  
        // ith factorial
        fact = (long)mulmod(fact, i, MOD);
  
        // product of first i factorials
        product = (long)mulmod(product, fact, MOD);
  
        // If at any iteration, product becomes
        // divisible by MOD, simply return 0;
        if (product == 0)
            return 0;
    }
    return product;
}
  
// Driver Code to Test above functions
public static void main(String[] args)
{
    long N = 3;
    System.out.println(findProduct(N));
  
    N = 5;
    System.out.println(findProduct(N));
  
}
}
// this Code is contributed by mits

Python3




# Python Program to find the
# product of first N factorials
  
# To compute (a * b) % MOD
def mulmod(a, b, mod):
    res = 0 # Initialize result
    a = a % mod
    while (b > 0):
  
        # If b is odd, add 'a' to result
        if (b % 2 == 1):
            res = (res + a) % mod
  
        # Multiply 'a' with 2
        a = (a * 2) % mod
  
        # Divide b by 2
        b //= 2
  
    # Return result
    return res % mod
  
# This function computes factorials and
# product by using above function i.e.
# modular multiplication
def findProduct(N):
    # Initialize product and fact with 1
    product = 1; fact = 1
    MOD = 1e9 + 7
    for i in range(1, N+1):
  
        # ith factorial
        fact = mulmod(fact, i, MOD)
  
        # product of first i factorials
        product = mulmod(product, fact, MOD)
  
        # If at any iteration, product becomes
        # divisible by MOD, simply return 0
        if not product:
            return 0
    return int(product)
  
# Driver Code to Test above functions
N = 3
print(findProduct(N))
N = 5
print(findProduct(N))
  
# This code is contributed by Ansu Kumari

C#




// C#  Program to find the
// product of first N factorials
  
using System;
  
public class GFG{
    // To compute (a * b) % MOD
static double mulmod(long a, long b, 
                                    long mod)
{
    long res = 0; // Initialize result
    a = a % mod;
    while (b > 0) {
  
        // If b is odd, add 'a' to result
        if (b % 2 == 1)
            res = (res + a) % mod;
  
        // Multiply 'a' with 2
        a = (a * 2) % mod;
  
        // Divide b by 2
        b /= 2;
    }
  
    // Return result
    return res % mod;
}
  
// This function computes factorials and
// product by using above function i.e.
// modular multiplication
static long findProduct(long N)
{
    // Initialize product and fact with 1
    long product = 1, fact = 1;
    long MOD = (long)(1e9 + 7);
    for (int i = 1; i <= N; i++) {
  
        // ith factorial
        fact = (long)mulmod(fact, i, MOD);
  
        // product of first i factorials
        product = (long)mulmod(product, fact, MOD);
  
        // If at any iteration, product becomes
        // divisible by MOD, simply return 0;
        if (product == 0)
            return 0;
    }
    return product;
}
  
// Driver Code to Test above functions
    static public void Main (){
        long N = 3;
        Console.WriteLine(findProduct(N));
        N = 5;
        Console.WriteLine(findProduct(N));
  
}
}
//This Code is contributed by ajit.

PHP




<?php
// PHP Program to find the
// product of first N factorials
  
// To compute (a * b) % MOD
function mulmod($a, $b, $mod)
{
    $res = 0; // Initialize result
    $a = $a % $mod;
    while ($b > 0) 
    {
  
        // If b is odd, add 'a' to result
        if ($b % 2 == 1)
            $res = ($res + $a) % $mod;
  
        // Multiply 'a' with 2
        $a = ($a * 2) % $mod;
  
        // Divide b by 2
        $b /= 2;
    }
  
    // Return result
    return $res % $mod;
}
  
// This function computes factorials and
// product by using above function i.e.
// modular multiplication
function findProduct($N)
{
    // Initialize product and fact with 1
    $product = 1;
    $fact = 1;
    $MOD = 1000000000;
    for ($i = 1; $i <= $N; $i++) 
    {
  
        // ith factorial
        $fact = mulmod($fact, $i, $MOD);
  
        // product of first i factorials
        $product = mulmod($product, $fact, $MOD);
  
        // If at any iteration, product becomes
        // divisible by MOD, simply return 0;
        if ($product == 0)
            return 0;
    }
    return $product;
}
  
// Driver Code
$N = 3;
echo findProduct($N),"\n";
  
$N = 5;
echo findProduct($N),"\n";
  
// This code is contributed by ajit
?>
Output:
12
34560

Time Complexity: O(N * logN), where O(log N) is the time complexity of modular multiplication.

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