Factorial of a large number - GeeksforGeeks

19 Jan, 2022
Factorial of a non-negative integer, is the 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.
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 a 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 the above algorithm. 
NOTE : In the below implementation, maximum digits in the output are assumed as 500. To find a factorial of a much larger number ( > 254), increase the size of an array or increase the value of MAX.
C++
Java
Python3
C#
PHP
Javascript
// 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 themvoid 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_sizeint 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;         // Store last digit of 'prod' in res[]         res[i] = prod % 10;          // Put rest in carry        carry  = prod/10;       }     // Put carry in res and increase result size    while (carry)    {        res[res_size] = carry%10;        carry = carry/10;        res_size++;    }    return res_size;} // Driver programint main(){    factorial(100);    return 0;}
// JAVA program to compute factorial// of big numbersclass 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 program to compute factorial# of big numbers import sys # This function finds factorial of large# numbers and prints themdef 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_sizedef 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[]        # make sure floor division is used        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        # make sure floor division is used        # to avoid floating value        carry = carry // 10        res_size = res_size + 1             return res_size # Driver programfactorial(100) #This code is contributed by Nikita Tiwari.
// C# program to compute// factorial of big numbersusing System; 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);         Console.WriteLine("Factorial of " +                       "given number is ");        for (int i = res_size - 1; i >= 0; i--)            Console.Write(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 Code    static public void Main ()    {                 factorial(100);    }} // This code is contributed by ajit
<?php// PHP program to compute factorial// of big numbers // Maximum number of digits in output$MAX = 500; // This function finds factorial of// large numbers and prints themfunction factorial($n){    global $MAX;    $res = array_fill(0, $MAX, 0);     // Initialize result    $res[0] = 1;    $res_size = 1;     // Apply simple factorial formula    // n! = 1 * 2 * 3 * 4...*n    for ($x = 2; $x <= $n; $x++)        $res_size = multiply($x, $res, $res_size);     echo "Factorial of given number is \n";    for ($i = $res_size - 1; $i >= 0; $i--)        echo $res[$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_sizefunction multiply($x, &$res, $res_size){    $carry = 0; // Initialize carry     // One by one multiply n with individual    // digits of res[]    for ($i = 0; $i < $res_size; $i++)    {        $prod = $res[$i] * $x + $carry;         // Store last digit of 'prod' in res[]        $res[$i] = $prod % 10;         // Put rest in carry        $carry = (int)($prod / 10);    }     // Put carry in res and increase    // result size    while ($carry)    {        $res[$res_size] = $carry % 10;        $carry = (int)($carry / 10);        $res_size++;    }    return $res_size;} // Driver Codefactorial(100);     // This code is contributed by chandan_jnu?>
<script> // Javascript program to compute factorial of big numbers // This function finds factorial of large numbers// and prints themfunction factorial(n){    let res = new Array(500);     // Initialize result    res[0] = 1;    let res_size = 1;     // Apply simple factorial formula n! = 1 * 2 * 3 * 4...*n    for (let x=2; x<=n; x++)        res_size = multiply(x, res, res_size);     document.write("Factorial of given number is " + "<br>");    for (let i=res_size-1; i>=0; i--)        document.write(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_sizefunction multiply(x, res, res_size){    let carry = 0; // Initialize carry     // One by one multiply n with individual digits of res[]    for (let i=0; i<res_size; i++)    {        let prod = res[i] * x + carry;         // Store last digit of 'prod' in res[]        res[i] = prod % 10;         // Put rest in carry        carry = Math.floor(prod/10);    }     // Put carry in res and increase result size    while (carry)    {        res[res_size] = carry%10;        carry = Math.floor(carry/10);        res_size++;    }    return res_size;} // Driver program    factorial(100); // This  code is contributed by Mayank Tyagi </script>
Factorial of given number is 
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
The above approach can be optimized in many ways. We will soon be discussing an optimized solution for the 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
Big Integer can also be used to calculate factorial of large numbers.
Java
// Java program to find large// factorials using BigIntegerimport java.math.BigInteger;import java.util.Scanner; public class Example {         // Returns Factorial of N    static BigInteger factorial(int N)    {        // Initialize result        BigInteger f            = new BigInteger("1"); // Or BigInteger.ONE         // Multiply f with 2, 3, ...N        for (int i = 2; i <= N; i++)            f = f.multiply(BigInteger.valueOf(i));         return f;    }     // Driver method    public static void main(String args[]) throws Exception    {        int N = 20;        System.out.println(factorial(N));    }}
2432902008176640000
Linked List can also be used, this approach will not waste any extra space.
C++
#include <bits/stdc++.h> using namespace std; #define rep(i, a, b) for (int i = a; i <= b; i++) using namespace std;// Made a class node containing data and previous pointer as// we are using tail pointerclass Node {public:    int data;    Node* prev;    Node(int n)    {        data = n;        prev = NULL;    }}; void Multiply(Node* tail, int n){    Node *temp = tail,         *prevNode = tail; // Temp variable for keeping tail    int carry = 0;    while (temp != NULL) {        int data = temp->data * n + carry;        temp->data = data % 10; // stores the last digit        carry = data / 10;        prevNode = temp;        temp = temp->prev; // Moving temp by 1 prevNode will                           // now denote temp    }    // If carry is greater than 0 then we create another    // node for it.    while (carry != 0) {        prevNode->prev = new Node((int)(carry % 10));        carry /= 10;        prevNode = prevNode->prev;    }} void print(Node* tail){    if (tail == NULL) // Using tail recursion        return;    print(tail->prev);    cout        << tail->data; // Print linked list in reverse order} // Driver codeint main(){    int n = 20;    Node tail(1); // Create a node and initialise it by 1    rep(i, 2, n)        Multiply(&tail, i); // Run a loop from 2 to n and                            // multiply with tail's i    print(&tail); // Print the linked list    cout << endl;    return 0;} // This code is contributed by Kingshuk Deb
2432902008176640000
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Morgan Stanley
Microsoft
MakeMyTrip
MAQ Software
Philips
BrowserStack
number-theory
Strings
Mathematical
Combinatorial
factorial
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