fix:Fit Binominal calculate.cpp to guidelines

math/binomial_calculate.cpp

@@ -3,44 +3,73 @@
  * @brief Program to calculate [Binomial
  * coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient)
  *
+ * @details
+ * quoted from: [Weisstein, Eric W. "Binomial Coefficient." From MathWorld--A
+ * Wolfram Web
+ * Resource.](https://mathworld.wolfram.com/BinomialCoefficient.html) "The
+ * binomial coefficient \f$ \binom{n}{k} \f$ is the number of ways of picking \f$k\f$ unordered
+ * outcomes from \f$n\f$ possibilities, also known as a combination or combinatorial
+ * number. The symbols \f$ _nC_k \f$ and \f$ \binom{n}{k} \f$ are used to
+ * denote a binomial coefficient, and are sometimes read as "n choose k." \f$ \binom{n}{k} \f$
+ * therefore gives the number of k-subsets possible out of a set of n distinct
+ * items. For example, The 2-subsets of {1,2,3,4} are the six pairs {1,2},
+ * {1,3}, {1,4}, {2,3}, {2,4}, and {3,4}, so \f$ \binom{4}{2} =6.\f$ " [Another good
+ * example/explanation](https://math.stackexchange.com/questions/2172355/probability-notation-two-numbers-stacked-inside-brackets)
+ *
+ * @note An identity of the binomial coefficient is \f$ \binom{n}{n-k} \f$ This is explained
+ * partially in the comments of this implementation but in more detail at [Prof.
+ * Tesler: Chapter 3.3, 4.1, 4.3. Binomial Coefficient
+ * Identities](https://mathweb.ucsd.edu/~gptesler/184a/slides/184a_ch4slides_17-handout.pdf
+ * page 2)
+ *
  * @author [astronmax](https://github.com/astronmax)
  */
 
 #include <cassert>   /// for assert
-#include <cstdint>   /// for int32_t type
-#include <cstdlib>   /// for atoi
 #include <iostream>  /// for IO operations
 
-/**
- * @namespace math
- * @brief Mathematical algorithms
- */
-namespace math {
 /**
  * @namespace binomial
  * @brief Functions for [Binomial
  * coefficients](https://en.wikipedia.org/wiki/Binomial_coefficient)
  * implementation
  */
-namespace binomial {
+namespace math { namespace binomial {
 /**
  * @brief Function to calculate binomial coefficients
- * @param n first value
- * @param k second value
- * @return binomial coefficient for n and k
+ * @param n number of possibilities
+ * @param k size of subset
+ * @return n if k == 1
+ * @return 1 if k == 0
+ * @return result if k != 0 || 1
  */
 size_t calculate(int32_t n, int32_t k) {
-    // basic cases
-    if (k > (n / 2))
-        k = n - k;
-    if (k == 1)
-        return n;
-    if (k == 0)
+    /*!
+     * Why do we do if k > (n/2) then k = n-k? Because \f$ \binom{n}{k} \f$ is the same as
+     * \f$ \binom{n}{n-k} \f$ or, in this case, \f$ \binom{6}{4} \f$ is the same as \f$ \binom{6}{2} \f$ since both are
+     * calculated for \f$ n!/k!(n-k)! \f$, in this case \f$ 6!/4!2! \f$. By replacing \f$ k \f$ with
+     * \f$ n-k \f$ we get \f$ 6!/2!4! \f$ which is the same sum. the benefit however, is our
+     * loop further on in this implementation now requires less iterations to
+     * find the same answer.
+     *
+     * k == 1 and k == 0 follow the same rule as \f$ n^1 \f$and \f$ n^0 \f$ respectively
+     */
+    if (k == 0) {
         return 1;
+    } else if (k == 1) {
+        return n;
+    } else if (k > (n / 2)) {
+        k = n - k;
+    }
 
+    /*!
+     * (n - k) + i returns 1 higher each loop imitating a factorial calculation.
+     * Then imagine i as the size of each set, divide by it to find the amount
+     * of possible sets in result amount of possibilities
+     */
     size_t result = 1;
     for (int32_t i = 1; i <= k; ++i) {
-        result *= n - k + i;
+        result *= (n - k) + i;
         result /= i;
     }
 
@@ -54,7 +83,6 @@ size_t calculate(int32_t n, int32_t k) {
  * @returns void
  */
 static void tests() {
-    // tests for calculate function
     assert(math::binomial::calculate(1, 1) == 1);
     assert(math::binomial::calculate(57, 57) == 1);
     assert(math::binomial::calculate(6, 3) == 20);
@@ -66,27 +94,15 @@ static void tests() {
     assert(math::binomial::calculate(60, 30) == 118264581564861424);
     assert(math::binomial::calculate(62, 31) == 465428353255261088);
 
-    std::cout << "[+] Binomial coefficients calculate test completed"
+    std::cout << "Binomial coefficients tests successfully passed"
               << std::endl;
 }
 
 /**
  * @brief Main function
- * @param argc commandline argument count
- * @param argv commandline array of arguments
  * @returns 0 on exit
  */
-int main(int argc, const char* argv[]) {
+int main() {
     tests();  // run self-test implementations
-
-    if (argc < 3) {
-        std::cout << "Usage ./binomial_calculate {n} {k}" << std::endl;
-        return 0;
-    }
-
-    int32_t n = atoi(argv[1]);
-    int32_t k = atoi(argv[2]);
-
-    std::cout << math::binomial::calculate(n, k) << std::endl;
     return 0;
 }