#include <stdio.h>

/**
 * @brief Computes the binomial coefficient "n choose k" (nCk).
 *
 * This function calculates the number of ways to choose k elements from a set
 * of n elements without repetition and without order. It uses an efficient
 * multiplicative approach to avoid large intermediate factorials.
 *
 * @param n The total number of elements.
 * @param k The number of elements to choose.
 * @return The computed binomial coefficient (n choose k), or 0 if k > n.
 */
unsigned long long binomial_coefficient(unsigned int n, unsigned int k);

unsigned long long binomial_coefficient(unsigned int n, unsigned int k) {
    if (k > n)
        return 0;
    if (k == 0 || k == n)
        return 1;

    if (k > n - k)
        k = n - k;

    unsigned long long result = 1;
    for (unsigned int i = 1; i <= k; ++i) {
        result *= n - (k - i);
        result /= i;
    }

    return result;
}

int main() {
    unsigned int n = 10, k = 3;
    printf("C(%u, %u) = %llu\n", n, k, binomial_coefficient(n, k));
    return 0;
}