feat: Add ncr mod p code (#1323)

math/ncr_modulo_p.cpp

@@ -0,0 +1,96 @@
+/**
+ * @file
+ * @brief This program aims at calculating nCr modulo p
+ *
+ */
+
+#include <cassert>
+#include <iostream>
+#include <vector>
+
+/** 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
+ */
+int64_t gcdExtended(int64_t a, int64_t b, int64_t *x, int64_t *y) {
+    if (a == 0) {
+        *x = 0, *y = 1;
+        return b;
+    }
+
+    int64_t x1 = 0, y1 = 0;
+    int64_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(int64_t a, int64_t m) {
+    int64_t x = 0, y = 0;
+    int64_t g = gcdExtended(a, m, &x, &y);
+    if (g != 1) {  // modular inverse doesn't exist
+        return -1;
+    }
+    else {
+        int64_t res = (x % m + m) % m;
+        return res;
+    }
+}
+
+std::vector<int64_t> fac;
+
+/** Find nCr % p
+ *
+ * @params[in] the numbers 'n', 'r' and 'p'
+ * @returns the value nCr % p
+ */
+int64_t ncr(int64_t n, int64_t r, int64_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;
+}
+
+int main() {
+    // populate the fac array
+    const int64_t size = 1e6 + 1;
+    fac = std::vector<int64_t>(size);
+    fac[0] = 1;
+    const int64_t p = 1e9 + 7;
+    for (int i = 1; i <= size; i++) {
+        fac[i] = (fac[i - 1] * i) % p;
+    }
+
+    // test 6Ci for i=0 to 7
+    for (int i = 0; i <= 7; i++) {
+        std::cout << 6 << "C" << i << " = " << ncr(6, i, p) << "\n";
+    }
+
+    // (52323 C 26161) % (1e9 + 7) = 224944353
+    assert(ncr(52323, 26161, p) == 224944353);
+    std::cout << "Assertion passed, (52323 C 26161) % (1e9 + 7) = 224944353\n";
+}