feat: Add ncr mod p code

DIRECTORY.md

@@ -143,6 +143,7 @@
   * [Least Common Multiple](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/least_common_multiple.cpp)
   * [Miller Rabin](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/miller_rabin.cpp)
   * [Modular Inverse Fermat Little Theorem](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/modular_inverse_fermat_little_theorem.cpp)
+  * [Ncr Modulo P](https://github.com/TheAlgorithms/C-Plus-Plus/blob/master/math/ncr_modulo_p.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/ncr_modulo_p.cpp

@@ -0,0 +1,138 @@
+/**
+ * @file
+ * @brief This program aims at calculating nCr modulo p
+ * @details nCr is defined as n! / (r! * (n-r)!) where n! represents factorial
+ * of n. In many cases, the value of nCr is too large to fit in a 64 bit
+ * integer. Hence, in competitive programming, there are many problems or
+ * subproblems to compute nCr modulo p where p is a given number.
+ * @author [Kaustubh Damania](https://github.com/KaustubhDamania)
+ */
+
+#include <cassert>   /// for assert
+#include <iostream>  /// for io operations
+#include <vector>    /// for std::vector
+
+/**
+ * @namespace math
+ * @brief Mathematical algorithms
+ */
+namespace math {
+/**
+ * @brief Class which contains all methods required for calculating nCr mod p
+ */
+class NCRModuloP {
+ private:
+    std::vector<uint64_t> fac;
+    uint64_t p;
+
+ public:
+    /** Constructor which precomputes the values of n! % mod from n=0 to size
+     *  and stores them in vector 'fac'
+     *  @params[in] the numbers 'size', 'mod'
+     */
+    NCRModuloP(uint64_t size, uint64_t mod) {
+        p = mod;
+        fac = std::vector<uint64_t>(size);
+        fac[0] = 1;
+        for (int i = 1; i <= size; i++) {
+            fac[i] = (fac[i - 1] * i) % p;
+        }
+    }
+
+    /** Finds the value of x, y such that a*x + b*y = gcd(a,b)
+     *
+     * @params[in] the numbers 'a', 'b' and address of 'x' and 'y' from above
+     * equation
+     * @returns the gcd of a and b
+     */
+    uint64_t gcdExtended(uint64_t a, uint64_t b, int64_t *x, int64_t *y) {
+        if (a == 0) {
+            *x = 0, *y = 1;
+            return b;
+        }
+
+        int64_t x1 = 0, y1 = 0;
+        uint64_t gcd = gcdExtended(b % a, a, &x1, &y1);
+
+        *x = y1 - (b / a) * x1;
+        *y = x1;
+        return gcd;
+    }
+
+    /** Find modular inverse of a with m i.e. a number x such that (a*x)%m = 1
+     *
+     * @params[in] the numbers 'a' and 'm' from above equation
+     * @returns the modular inverse of a
+     */
+    int64_t modInverse(uint64_t a, uint64_t m) {
+        int64_t x = 0, y = 0;
+        uint64_t g = gcdExtended(a, m, &x, &y);
+        if (g != 1) {  // modular inverse doesn't exist
+            return -1;
+        } else {
+            int64_t res = ((x + m) % m);
+            return res;
+        }
+    }
+
+    /** Find nCr % p
+     *
+     * @params[in] the numbers 'n', 'r' and 'p'
+     * @returns the value nCr % p
+     */
+    int64_t ncr(uint64_t n, uint64_t r, uint64_t p) {
+        // Base cases
+        if (r > n) {
+            return 0;
+        }
+        if (r == 1) {
+            return n % p;
+        }
+        if (r == 0 || r == n) {
+            return 1;
+        }
+        // fac is a global array with fac[r] = (r! % p)
+        int64_t denominator = modInverse(fac[r], p);
+        if (denominator < 0) {  // modular inverse doesn't exist
+            return -1;
+        }
+        denominator = (denominator * modInverse(fac[n - r], p)) % p;
+        if (denominator < 0) {  // modular inverse doesn't exist
+            return -1;
+        }
+        return (fac[n] * denominator) % p;
+    }
+};
+}  // namespace math
+
+/**
+ * @brief Test implementations
+ * @param ncrObj object which contains the precomputed factorial values and
+ * ncr function
+ * @returns void
+ */
+void tests(math::NCRModuloP ncrObj) {
+    // (52323 C 26161) % (1e9 + 7) = 224944353
+    assert(ncrObj.ncr(52323, 26161, 1000000007) == 224944353);
+    // 6 C 2 = 30, 30%5 = 0
+    assert(ncrObj.ncr(6, 2, 5) == 0);
+    // 7C3 = 35, 35 % 29 = 8
+    assert(ncrObj.ncr(7, 3, 29) == 6);
+}
+
+/**
+ * @brief Main function
+ * @returns 0 on exit
+ */
+int main() {
+    // populate the fac array
+    const uint64_t size = 1e6 + 1;
+    const uint64_t p = 1e9 + 7;
+    math::NCRModuloP ncrObj = math::NCRModuloP(size, p);
+    // test 6Ci for i=0 to 7
+    for (int i = 0; i <= 7; i++) {
+        std::cout << 6 << "C" << i << " = " << ncrObj.ncr(6, i, p) << "\n";
+    }
+    tests(ncrObj);  // execute the tests
+    std::cout << "Assertions passed\n";
+}