from fractions import Fraction
from itertools import combinations


def mul(xs):
    r = 1
    for x in xs:
        r *= x
    return r


def mul_inv(xs):
    r = 1
    for x in xs:
        r *= (1 - x)
    return r


def sub_odds(n_blue, n_total):
    """ Computes the odds for getting exactly n_blue out of n_total blue discs.
    """
    total_odds = 0
    odds = [Fraction(1, n) for n in range(2, 2 + n_total)]
    odds_set = set(odds)
    for odd in combinations(odds, n_blue):
        odd_inv = tuple(odds_set.difference(set(odd)))
        total_odds += mul(odd) * mul_inv(odd_inv)
    return total_odds


def euler_121():
    n_turns = 15
    n_to_win = n_turns // 2 + 1
    odds = 0
    for n_blue in range(n_to_win, n_turns + 1):
        odds += sub_odds(n_blue, n_turns)
    price = int(1 / odds)
    return price


if __name__ == "__main__":
    solution = euler_121()
    print("e121.py: " + str(solution))
    assert(solution == 2269)