Factorial

Factorial is a vital function in Mathematics, which plays a significant role in arranging or ordering sets of numbers. This concept, discovered by Daniel Bernoulli, is utilized in various mathematical areas such as probability, permutations and combinations, sequences, and series. In simple terms, a factorial is a function that multiplies a number by every number below it until you reach 1. For instance, the factorial of 3 or 3! equals 3 × 2 × 1, which equals 6. This article provides a comprehensive understanding of the factorial, its notation, formula, examples and more.Article Contents: What is Factorial? Factorial Notation Factorial Formula Calculating Factorial of a Number Factorial of 10 Factorial Table for Numbers 1 to 10 What is Sub Factorial? Factorial of 5 Factorial Examples Practice Problems Recommended Tool: Factorial Calculator Defining FactorialIn Mathematics, factorial is a simple yet powerful concept. Factorials are just products, indicated by an exclamation mark. Factorial is the multiplication of a number with all the natural numbers that are less than it. Let's delve deeper into the definition, formula, and examples of factorial. Formula for FactorialThe formula to calculate the factorial of a number is as follows:n! = n × (n-1) × (n-2) × (n-3) × ….× 3 × 2 × 1 For an integer n ≥ 1, the factorial representation in terms of pi product notation is:\(\begin{array}{l}n! = \prod_{i=1}^{n}i\end{array} \)From the above formulas, the recurrence relation for the factorial of a number is defined as the product of the factorial number and factorial of

A number through C ProgramFactorial of a number means the product of all positive descending integers. Factorial is denoted by ‘!’.Example:-5! = 5*4*3*2*1 = 1204! = 4*3*2*1 = 24There are 2 ways to write the program for finding the factorial of a number.Finding factorial using loopFinding factorial using recursionFind factorial of a number using loop:-In C, you make use of loops to find the factorial of a number.#include int main() { int a,factorial=1,num; printf("TechVidvan Tutorial: Find the factorial of a number using a loop!\n"); printf("Please enter the number: "); scanf("%d",#); for(a=1;aOutput:-TechVidvan Tutorial: Find the factorial of a number using a loop!Please enter the number: 5The factorial value of 5 is: 120Find factorial of a number using recursion:-In C, you also calculate the factorial of a number using recursion.#include int fact(int num) { if (num == 0) return 1; else return(num * fact(num-1)); } int main() { int num; int factorial; printf("TechVidvan Tutorial: Finding the factorial of a number using recursion!\n"); printf("Please enter a number: "); scanf("%d", #); factorial = fact(num); printf("The factorial value of %d is %d\n", num, factorial); return 0; }Output:-TechVidvan Tutorial: Finding the factorial of a number using recursion!Please enter a number: 5The factorial value of 5 is 1205. Reverse a number using C:-In C, there are many ways to reverse a number. You can reverse a number using loops and recursion.Algorithm:-You have to use modulus(%) to reverse a number.First, you have to initialize a reverse number to 0.Then, multiply the reverse with value 10.After that, divide the

That number minus 1. It is given by:n! = n. (n-1) ! Calculating Factorial of a Number To calculate the factorial of any given number, you simply substitute the value for n in the above formula. The expansion of the formula gives the numbers to be multiplied together to get the factorial of the number. Computing Factorial of 10 For example, the factorial of 10 can be calculated as follows:10! = 10. 9 !10! = 10 (9 × 8 × 7 × 6 × 5× 4 × 3 × 2 × 1)10! = 10 (362,880)10! = 3,628,800Therefore, the factorial of 10 is 3,628,800 The factorial function is widely used in various fields of Mathematics such as algebra, permutation and combination , and mathematical analysis. Its main application is to count the possible distinct arrangements of “n” objects.For instance, the number of ways in which 4 persons can be seated in a row can be found using the factorial. That means, the factorial of 4 gives the required number of ways, i.e. 4! = 4 × 3 × 2 × 1 = 24. Hence, 4 persons can be seated in a row in 24 ways. Factorial Table for Numbers 1 to 10Here is a list of factorial values for numbers 1 to 10: n Factorial of a Number (n!) Expansion Value 1 1! 1 1 2 2! 2 × 1 2 3 3! 3 × 2 × 1 6 4 4! 4 × 3 × 2 × 1 24 5 5!

5 × 4 × 3 × 2 × 1 120 6 6! 6 × 5 × 4 × 3 × 2 × 1 720 7 7! 7 × 6 × 5 × 4 × 3 × 2 × 1 5,040 8 8! 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 40,320 9 9! 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 362,880 10 10! 10 × 9 ×8 × 7 × 6 × 5 ×4 × 3 × 2 × 1 3,628,800 Understanding Sub FactorialSub factorial, represented by “!n”, is a mathematical term defined as the number of rearrangements of n objects. It refers to the number of permutations of n objects where no object is in its original position. The formula to calculate the sub-factorial of a number is as follows:\(\begin{array}{l}!n = n!\sum_{k=0}^{n}\frac{(-1)^{k}}{k!}\end{array} \)Calculating Factorial of 5 Calculating the factorial of 5 is straightforward. You can find it by using the formula and expanding the numbers. Let's see how it's done. We know that, n! = 1 × 2 × 3 …… × n Factorial of 5 can be calculated as follows: 5! = 1 × 2 × 3 × 4 × 5 5! = 120 Therefore, the factorial of 5 equals 120. Factorial ExamplesExample 1: What is the factorial of 6? Solution: We know that the factorial formula is n! = n × (n – 1) × (n – 2) × (n

Than or equal to 1, So it will return 1.Unwinding the StackNow, the recursive calls will start returning: After the 4th call now it will again start from back, returning to the Third Call first.Return to Third Call: factorial(2)We already have factorial(1) = 1, therefor factorial(2) will return, "2 * factorial(1)", that’s "2 * 1" , which returns as factorial(2) equals to 2.Return to Second Call: factorial(3)Now, factorial(2) is 2, therefore factorial(3) equals to "3 * 2", that’s 6.Return to Initial Call: factorial(4)We have factoria(3) which returns 6, therefore, factorial(4) returns "4 * 6 = 24".Types of RecursionRecursion can be categorized into two main types where each with its own sub-categories −1. Direct RecursionDirect recursion occurs when a function calls itself directly −Simple Direct RecursionThe function calls itself with a simpler or smaller instance of the problem. It is used for solving problems like factorial calculation, fibonacci sequence generation, etc.Tail RecursionA form of direct recursion where the recursive call is the last operation in the function. It is used for solving accumulative calculations and list processing problems.int factorial(int n, int result = 1) { if (n Head RecursionThe recursive call is made before any other operation in the function. Processing occurs after the recursive call returns. It is used for tree traversals and output generation.void printNumbers(int n) { if (n > 0) { printNumbers(n - 1); // Recursive call first cout Linear RecursionEach function call generates exactly one recursive call, forming a linear chain of calls. It is used for simple counting or summing.int linearRecursion(int n) { if (n 2. Indirect RecursionIndirect recursion occurs when a function calls another function, which eventually leads to the original function being called. This involves two or more functions calling each other.Mutual RecursionIn mutual recursion, two or more functions call each other in a recursive manner, forming a cyclic dependency. It is used for even and odd number classification and grammar parsing.#include using namespace std;void even(int n);void odd(int n);void even(int n) { if (n == 0) { cout OutputEvenEvenNested RecursionThe nested recursion is a form of indirect recursion where a recursive function makes another recursive call inside its own recursive call. It is used for solving complex mathematical and algorithmic problems.#include using namespace std;int nestedRecursion(int n) { if (n > 100) { return n - 10; } else { return nestedRecursion(nestedRecursion(n + 11)); // Nested recursive calls }}int main() { cout Output91Advantages of RecursionSimplicity

Recursion is a programming technique where a function calls itself over again and again with modified arguments until it reaches its base case, where the recursion stops.It breaks a problem down into smaller, more manageable sub-problems, recursion allows for elegant and better solutions to complex problems.Recursive FunctionA recursive function is a function which is particularly used for recursion where a function calls itself, either directly or indirectly, to address a problem. It must include at least one base case to terminate the recursion and one recursive case where the function invokes itself.Base Case ��� It’s a case where recursion stops or ends after reaching that particular condition.Recursive Case − It’s a case where a function calls itself over again with decremented value until and unless it reaches its base case.Creating a Recursive FunctionThe following syntax is used to implement a recursive function in C++ −function name(param_1, param_2..){ }Here,Where, function name(param_1, param_2..) is a function declared as "name" passing with multiple parameters in it as per requirement.Now the function body is being divided into three sub-categories : base condition, function body and return statement.In base condition we will define its base case, where recursion has to stop or end.In the function body, a recursive case will be defined, where we need to call the function over again and again as per requirement.At last, the return statement will return the final output of the function.Calling a Recursive FunctionCalling a recursive function is just like calling any other function, where you will use the function's name and provide the necessary parameters in int main() body.To call a recursive function, use the following syntax −func_name(value);Example of RecursionBelow is an example of a recursion function in C++. Here, we are calculating the factorial of a number using the recursion −#include using namespace std;// Recursive Function to Calculate Factorialint factorial(int num) { // Base case if (num > positive_number; if (positive_number OutputEnter a positive integer: 4 (input)Factorial of 4 is 24ExplanationIf take input int positive_number as 4, It will send integer to function name 'factorial' as factorial(4)Initial Call: factorial(4)This function will check base case (nSecond call: factorial(3)This function will again check base case, as it’s not satisfying it, therefor will again move forward to recursive case , and compute as "3 * factorial(2)".Third Call: factorial(2)Checks base case and computes "2 * factorial(1)"Fourth call: factorial(1)Checks base case, now since function satisfying this base case condition that’s less