/* There are exactly ten ways of selecting three from five, 12345:
 *
 * 123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
 *
 * In combinatorics, we use the notation, (5 3) = 10.
 *
 * In general, (n r)=n!/(r!(n−r)!), where r≤n, n!=n×(n−1)×...×3×2×1, and 0!=1.
 *
 * It is not until n=23, that a value exceeds one-million: (23 10)=1144066.
 *
 * How many, not necessarily distinct, values of (n r) for 1≤n≤100, are greater than one-million?*/

#define _POSIX_C_SOURCE 199309L

#include <stdio.h>
#include <stdlib.h>
#include <time.h>
#include <gmp.h>

#define N 100
#define LIMIT 1000000

void binomial(mpz_t bin, unsigned long int n, unsigned long int k, mpz_t *factorials);

int main(int argc, char **argv)
{
    int i, j, count = 0;
    double elapsed;
    struct timespec start, end;
    mpz_t bin;
    mpz_t factorials[N+1];

    clock_gettime(CLOCK_MONOTONIC, &start);

    mpz_init(bin);

    mpz_init_set_ui(factorials[0], 1);

    /* Straightforward brute force approach, using the GMP Library. First
     * calculate all factorials, then use them for the binomial coefficients.*/
    for(i = 1; i <= N; i++)
    {
        mpz_init(factorials[i]);
        mpz_fac_ui(factorials[i], i);
    }

    for(i = 23; i <= N; i++)
    {
        for(j = 1; j <= i; j++)
        {
            binomial(bin, i, j, factorials);

            if(mpz_cmp_ui(bin, LIMIT) > 0)
                count++;
        }
    }

    mpz_clear(bin);

    clock_gettime(CLOCK_MONOTONIC, &end);

    elapsed = (end.tv_sec - start.tv_sec) + (double)(end.tv_nsec - start.tv_nsec) / 1000000000;

    printf("Project Euler, Problem 53\n");
    printf("Answer: %d\n", count);

    printf("Elapsed time: %.9lf seconds\n", elapsed);

    return 0;
}

void binomial(mpz_t bin, unsigned long n, unsigned long int k, mpz_t *factorials)
{
    mpz_t num, d1, d2;

    mpz_inits(num, d1, d2, NULL);

    mpz_set(num, factorials[n]);
    mpz_set(d1, factorials[k]);
    mpz_set(d2, factorials[n-k]);

    mpz_mul(d1, d1, d2);
    mpz_tdiv_q(bin, num, d1);

    mpz_clears(num, d1, d2, NULL);
}