**Understanding Factorials**

**Definition:**The factorial of a non-negative integer n, denoted as n!, is the product of all positive integers less than or equal to n.**Representation:**Mathematically, n! = n * (n-1) * (n-2) * … * 2 * 1.**Examples:**- 5! = 5 * 4 * 3 * 2 * 1 = 120
- 0! = 1 (by definition)

**Key Concepts for Implementation**

**Functions:**C programs use functions to organize code and make it reusable.**Iterative Approach:**Factorials can be calculated using a loop that iterates from 1 to n, multiplying the numbers together.**Recursive Approach:**Factorials can also be calculated recursively, where a function calls itself with a smaller value of n until a base case is reached.

**Implementation Using a Function (Iterative Approach)**

**Code:**

C

`#`**include** <stdio.h>
long long factorial(int n) {
long long result = 1; // Initialize result to 1
for (int i = 1; i <= n; i++) {
result *= i; // Multiply result by each number from 1 to n
}
return result; // Return the calculated factorial
}
int main() {
int num;
printf("Enter a non-negative number: ");
scanf("%d", &num);
if (num < 0) {
printf("Factorial is not defined for negative numbers.\n");
} else {
long long fact = factorial(num);
printf("Factorial of %d = %lld\n", num, fact);
}
return 0;
}

**Explanation:**

**Header:**`#include <stdio.h>`

includes the standard input/output library for functions like`printf`

and`scanf`

.**Function**`factorial`

:- Takes an integer
`n`

as input. - Initializes
`result`

to 1. - Uses a
`for`

loop to iterate from 1 to`n`

. - In each iteration, multiplies
`result`

by`i`

. - Returns the calculated
`result`

.

- Takes an integer
**Main Function:**- Prompts the user to enter a number.
- Checks if the number is non-negative.
- Calls the
`factorial`

function to calculate the factorial. - Prints the result.

**Implementation Using a Function (Recursive Approach)**

**Code:**

C

`#`**include** <stdio.h>
long long factorial(int n) {
if (n == 0) {
return 1; // Base case: factorial of 0 is 1
} else {
return n * factorial(n - 1); // Recursive call
}
}
// Rest of the main function remains the same

**Explanation:**

- The recursive approach breaks down the problem into smaller, self-similar subproblems.
- The base case stops the recursion when
`n`

reaches 0. - The recursive case calls
`factorial(n - 1)`

to calculate the factorial of a smaller number.

**Additional Considerations**

**Data Type:**Choose an appropriate data type (like`long long`

) to accommodate larger factorial values.**Error Handling:**Consider handling invalid input (e.g., negative numbers) gracefully.**Optimization:**Explore iterative and recursive approaches for efficiency based on specific use cases.