/* Starting in the top left corner of a 2×2 grid, and only being able to move to the right and down, there are exactly 6 routes to the bottom right corner
 * How many such routes are there through a 20×20 grid?*/

#define _POSIX_C_SOURCE 199309L

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

int main(int argc, char **argv)
{
    double elapsed;
    struct timespec start, end;
    mpz_t count, tmp;

    clock_gettime(CLOCK_MONOTONIC, &start);

    /* Using a combinatorial solution: in a 20x20 grid there will always be
     * 20 movements to the right and 20 movements down, that can be represented
     * as a string of Rs and Ds. The number of routes is the number of combinations.
     * This is obtained calculating n!/(k!*(n-k)!), where n=40 and k=20. The GMP
     * Library is used to calculate the factorials.*/
    mpz_inits(count, tmp, NULL);
    mpz_fac_ui(count, 40);
    mpz_fac_ui(tmp, 20);
    mpz_mul(tmp, tmp, tmp);
    mpz_tdiv_q(count, count, tmp);

    clock_gettime(CLOCK_MONOTONIC, &end);

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

    printf("Project Euler, Problem 15\n");
    gmp_printf("Answer: %Zd\n", count);

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

    mpz_clears(count, tmp, NULL);

    return 0;
}