Factorial of a large number

2.9

Factorial of a non-negative integer, is multiplication of all integers smaller than or equal to n. For example factorial of 6 is 6*5*4*3*2*1 which is 720.

factorial

We have discussed simple program for factorial.

How to compute factorial of 100 using a C/C++ program?
Factorial of 100 has 158 digits. It is not possible to store these many digits even if we use long long int.

Examples:

Input : 100
Output : 933262154439441526816992388562667004-
         907159682643816214685929638952175999-
         932299156089414639761565182862536979-
         208272237582511852109168640000000000-
         00000000000000

Input :50
Output : 3041409320171337804361260816606476884-
         4377641568960512000000000000

Following is a simple solution where we use an array to store individual digits of the result. The idea is to use basic mathematics for multiplication.

The following is detailed algorithm for finding factorial.

factorial(n)
1) Create an array ‘res[]’ of MAX size where MAX is number of maximum digits in output.
2) Initialize value stored in ‘res[]’ as 1 and initialize ‘res_size’ (size of ‘res[]’) as 1.
3) Do following for all numbers from x = 2 to n.
……a) Multiply x with res[] and update res[] and res_size to store the multiplication result.

How to multiply a number ‘x’ with the number stored in res[]?
The idea is to use simple school mathematics. We one by one multiply x with every digit of res[]. The important point to note here is digits are multiplied from rightmost digit to leftmost digit. If we store digits in same order in res[], then it becomes difficult to update res[] without extra space. That is why res[] is maintained in reverse way, i.e., digits from right to left are stored.

multiply(res[], x)
1) Initialize carry as 0.
2) Do following for i = 0 to res_size – 1
….a) Find value of res[i] * x + carry. Let this value be prod.
….b) Update res[i] by storing last digit of prod in it.
….c) Update carry by storing remaining digits in carry.
3) Put all digits of carry in res[] and increase res_size by number of digits in carry.

Example to show working of multiply(res[], x)
A number 5189 is stored in res[] as following.
res[] = {9, 8, 1, 5}
x = 10

Initialize carry = 0;

i = 0, prod = res[0]*x + carry = 9*10 + 0 = 90.
res[0] = 0, carry = 9

i = 1, prod = res[1]*x + carry = 8*10 + 9 = 89
res[1] = 9, carry = 8

i = 2, prod = res[2]*x + carry = 1*10 + 8 = 18
res[2] = 8, carry = 1

i = 3, prod = res[3]*x + carry = 5*10 + 1 = 51
res[3] = 1, carry = 5

res[4] = carry = 5

res[] = {0, 9, 8, 1, 5} 

Below is the implementation of above algorithm.

C++

// C++ program to compute factorial of big numbers
#include<iostream>
using namespace std;

// Maximum number of digits in output
#define MAX 500

int multiply(int x, int res[], int res_size)

// This function finds factorial of large numbers and prints them
void factorial(int n)
{
    int res[MAX];

    // Initialize result
    res[0] = 1;
    int res_size = 1;

    // Apply simple factorial formula n! = 1 * 2 * 3 * 4...*n
    for (int x=2; x<=n; x++)
        res_size = multiply(x, res, res_size);

    cout << "Factorial of given number is n";
    for (int i=res_size-1; i>=0; i--)
        cout << res[i];
}

// This function multiplies x with the number represented by res[].
// res_size is size of res[] or number of digits in the number represented
// by res[]. This function uses simple school mathematics for multiplication.
// This function may value of res_size and returns the new value of res_size
int multiply(int x, int res[], int res_size)
{
    int carry = 0;  // Initialize carry

    // One by one multiply n with individual digits of res[]
    for (int i=0; i<res_size; i++)
    {
        int prod = res[i] * x + carry;
        res[i] = prod % 10;  // Store last digit of 'prod' in res[]
        carry  = prod/10;    // Put rest in carry
    }

    // Put carry in res and increase result size
    while (carry)
    {
        res[res_size] = carry%10;
        carry = carry/10;
        res_size++;
    }
    return res_size;
}

// Driver program
int main()
{
    factorial(100);
    return 0;
}

Java

// JAVA program to compute factorial 
// of big numbers
class GFG {
    
    // This function finds factorial of 
    // large numbers and prints them
    static void factorial(int n)
    {
        int res[] = new int[500];

        // Initialize result
        res[0] = 1;
        int res_size = 1;

        // Apply simple factorial formula 
        // n! = 1 * 2 * 3 * 4...*n
        for (int x = 2; x <= n; x++)
            res_size = multiply(x, res, res_size);

        System.out.println("Factorial of given number is ");
        for (int i = res_size - 1; i >= 0; i--)
            System.out.print(res[i]);
    }
    
    // This function multiplies x with the number
    // represented by res[]. res_size is size of res[] or 
    // number of digits in the number represented by res[].
    // This function uses simple school mathematics for 
    // multiplication. This function may value of res_size
    // and returns the new value of res_size
    static int multiply(int x, int res[], int res_size)
    {
        int carry = 0; // Initialize carry

        // One by one multiply n with individual 
        // digits of res[]
        for (int i = 0; i < res_size; i++)
        {
            int prod = res[i] * x + carry;
            res[i] = prod % 10; // Store last digit of 
                                // 'prod' in res[]
            carry = prod/10; // Put rest in carry
        }

        // Put carry in res and increase result size
        while (carry!=0)
        {
            res[res_size] = carry % 10;
            carry = carry / 10;
            res_size++;
        }
        return res_size;
    }

    // Driver program
    public static void main(String args[])
    {
        factorial(100);
    }
}
//This code is contributed by Nikita Tiwari

Python

# Python program to compute factorial
# of big numbers

import sys

# This function finds factorial of large
# numbers and prints them
def factorial( n) :
    res = [None]*500
    # Initialize result
    res[0] = 1
    res_size = 1

    # Apply simple factorial formula 
    # n! = 1 * 2 * 3 * 4...*n
    x = 2
    while x <= n :
        res_size = multiply(x, res, res_size)
        x = x + 1
    
    print ("Factorial of given number is")
    i = res_size-1
    while i >= 0 :
        sys.stdout.write(str(res[i]))
        sys.stdout.flush()
        i = i - 1
        

# This function multiplies x with the number 
# represented by res[]. res_size is size of res[] 
# or number of digits in the number represented 
# by res[]. This function uses simple school 
# mathematics for multiplication. This function 
# may value of res_size and returns the new value
# of res_size
def multiply(x, res,res_size) :
    
    carry = 0 # Initialize carry

    # One by one multiply n with individual
    # digits of res[]
    i = 0
    while i < res_size :
        prod = res[i] *x + carry
        res[i] = prod % 10; # Store last digit of 
                            # 'prod' in res[]
        carry = prod/10; # Put rest in carry
        i = i + 1

    # Put carry in res and increase result size
    while (carry) :
        res[res_size] = carry % 10
        carry = carry / 10
        res_size = res_size + 1
        
    return res_size

# Driver program
factorial(100)

#This code is contributed by Nikita Tiwari.


Output:
Factorial of given number is
9332621544394415268169923885626670049071596826438162146859296389
5217599993229915608941463976156518286253697920827223758251185210
916864000000000000000000000000

The above approach can be optimized in many ways. We will soon be discussing optimized solution for same.

This article is contributed by Harshit Agrawal. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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