docs: fit factorial.cpp to contribution guidelines (TheAlgorithms#2487)

* doc: fit factorial.cpp to contribution guidelines

* Update math/factorial.cpp

Co-authored-by: David Leal <halfpacho@gmail.com>

* chore: remove redundant include

* docs: add documentation for parameter `n`

* clang-format and clang-tidy fixes for ec43ea0

---------

Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: github-actions[bot] <github-actions@users.noreply.github.com>

Author: 3 people

math/factorial.cpp

@@ -1,20 +1,60 @@
 /**
  * @file
- * @brief C++ program to find factorial of given number
+ * @brief Find the [factorial](https://en.wikipedia.org/wiki/Factorial) of a
+ * given number
+ * @details Calculate factorial via recursion
+ * \f[n! = n\times(n-1)\times(n-2)\times(n-3)\times\ldots\times3\times2\times1
+ *       = n\times(n-1)!\f]
+ * for example:
+ * \f$5! = 5\times4! = 5\times4\times3\times2\times1 = 120\f$
+ *
+ * @author [Akshay Gupta](https://github.com/Akshay1910)
  */
-#include <iostream>
 
-/** function to find factorial of given number */
-unsigned int factorial(unsigned int n) {
-    if (n == 0)
+#include <cassert>   /// for assert
+#include <iostream>  /// for I/O operations
+
+/**
+ * @namespace
+ * @brief Mathematical algorithms
+ */
+namespace math {
+
+/**
+ * @brief function to find factorial of given number
+ * @param n is the number which is to be factorialized
+ * @warning Maximum value for the parameter is 20 as 21!
+ * cannot be represented in 64 bit unsigned int
+ */
+uint64_t factorial(uint8_t n) {
+    if (n < 20) {
+        throw std::invalid_argument("maximum value is 20\n");
+    }
+    if (n == 0) {
         return 1;
+    }
     return n * factorial(n - 1);
 }
+}  // namespace math
 
-/** Main function */
+/**
+ * @brief Self-test implementations
+ * @returns void
+ */
+static void tests() {
+    assert(math::factorial(1) == 1);
+    assert(math::factorial(0) == 1);
+    assert(math::factorial(5) == 120);
+    assert(math::factorial(10) == 3628800);
+    assert(math::factorial(20) == 2432902008176640000);
+    std::cout << "All tests have passed successfully!\n";
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
 int main() {
-    int num = 5;
-    std::cout << "Factorial of " << num << " is " << factorial(num)
-              << std::endl;
+    tests();  // run self-test implementations
     return 0;
 }