Feat: added n_choose_r combinatorics algorithm (#1429)

* feat: added n_choose_r combinatorial algorithm

* fix: some type casting issue

* fix: MathJax documentation fix

* fix: MathJax documentation fix

* fix: MathJax notation fix

* fix: header comments

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

* fix: suggested changes to documentation

* updating DIRECTORY.md

* fix: suggested changes

* fix: suggested changes

* fix: namespace closing comment

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

* fix: removing a blank line

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

* fix: link issue

* fix: removed link

* fix: removed newline

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

Co-authored-by: David Leal <halfpacho@gmail.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

DIRECTORY.md

@@ -153,6 +153,7 @@
   * [Miller Rabin](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/miller_rabin.cpp)
   * [Modular Exponentiation](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_exponentiation.cpp)
   * [Modular Inverse Fermat Little Theorem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_inverse_fermat_little_theorem.cpp)
+  * [N Choose R](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/n_choose_r.cpp)
   * [Number Of Positive Divisors](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/number_of_positive_divisors.cpp)
   * [Power For Huge Numbers](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/power_for_huge_numbers.cpp)
   * [Prime Factorization](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/prime_factorization.cpp)

math/n_choose_r.cpp

@@ -0,0 +1,81 @@
+/**
+ * @file
+ * @brief [Combinations](https://en.wikipedia.org/wiki/Combination) n choose r function implementation
+ * @details
+ * A very basic and efficient method of calculating
+ * choosing r from n different choices.
+ * \f$ \binom{n}{r} = \frac{n!}{r! (n-r)!} \f$
+ *
+ * @author [Tajmeet Singh](https://github.com/tjgurwara99)
+ */
+
+#include <iostream>   /// for io operations
+#include <cassert>   /// for assert
+
+/**
+ * @namespace math
+ * @brief Mathematical algorithms
+ */
+namespace math {
+/**
+ * @brief This is the function implementation of \f$ \binom{n}{r} \f$
+ * @details
+ * We are calculating the ans with iterations
+ * instead of calculating three different factorials.
+ * Also, we are using the fact that 
+ * \f$ \frac{n!}{r! (n-r)!} = \frac{(n - r + 1) \times \cdots \times n}{1 \times \cdots \times r} \f$
+ * @tparam T Only for integer types such as long, int_64 etc
+ * @param n \f$ n \f$ in \f$ \binom{n}{r} \f$
+ * @param r \f$ r \f$ in \f$ \binom{n}{r} \f$
+ * @returns ans \f$ \binom{n}{r} \f$
+ */
+template <class T>
+T n_choose_r(T n, T r) {
+    if(r > n / 2)
+        r = n - r; // Because of the fact that  nCr(n, r) == nCr(n, n - r)
+    T ans = 1;
+    for(int i = 1; i <= r; i++) {
+        ans *= n - r + i;
+        ans /= i;
+    }
+    return ans;
+}
+} // namespace math
+
+/**
+ * @brief Test implementations
+ * @returns void
+ */
+static void test() {
+    // First test on 5 choose 2
+     uint8_t t = math::n_choose_r(5, 2);
+     assert(((void)"10 is the answer but function says otherwise.\n",
+             t == 10));
+     std::cout << "First test passes." << std::endl;
+     // Second test on 5 choose 3
+     t = math::n_choose_r(5, 3);
+     assert(((void)"10 is the answer but the function says otherwise.\n",
+             t == 10));
+     std::cout << "Second test passes." << std::endl;
+     // Third test on 3 choose 2
+     t = math::n_choose_r(3, 2);
+     assert(((void)"3 is the answer but the function says otherwise.\n",
+             t == 3));
+     std::cout << "Third test passes." << std::endl;
+     // Fourth test on 10 choose 4
+     t = math::n_choose_r(10, 4);
+     assert(((void)"210 is the answer but the function says otherwise.\n",
+             t == 210));
+     std::cout << "Fourth test passes." << std::endl;
+}
+
+/**
+ * @brief Main function
+ * @param argc commandline argument count (ignored)
+ * @param argv commandline array of arguments (ignored)
+ * @returns 0 on exit
+ */
+int main(int argc, char *argv[]) {
+    test(); // executing tests
+    return 0;
+}