from fractions import Fraction


def get_probability_1(n=10, s=5):
    total = 0
    successes = 0
    bag = frozenset(range(1, n+1))
    for a in bag:
        bag_a = bag - {a}
        for b in bag_a:
            bag_b = bag_a - {b}
            for c in bag_b:
                total += 1
                if b+c == s:
                    successes += 1
                    print(a, b, c)
    return Fraction(successes, total)


def get_probability_2(n=10, s=5):
    if s % 2 == 1:
        return Fraction(n*(s-1) - 2*(s-1), n*(n-1)*(n-2))
    else:
        return Fraction(n*(s-2) - 2*(s-2), n*(n-1)*(n-2))


def get_probability_3(n=10, s=5):
    return Fraction(s - 2 + s % 2, n*(n-1))


if __name__ == "__main__":
    n, s = 10, 5
    print(get_probability_1(n, s))
    print(get_probability_2(n, s))
    print(get_probability_3(n, s))

from fractions import Fraction


def get_probability_1(n=10, s=5):
    total = 0
    successes = 0
    bag = frozenset(range(1, n+1))
    for a in bag:
        bag_a = bag - {a}
        for b in bag_a:
            bag_b = bag_a - {b}
            for c in bag_b:
                total += 1
                if b+c == s:
                    successes += 1
                    print(a, b, c)
    return Fraction(successes, total)


def get_probability_2(n=10, s=5):
    c = max(0, min(n, s - 1) - max(1, s - n) + s % 2)  # Number of combinations with sum s
    return Fraction(n * c - 2 * c, n * (n - 1) * (n - 2))


def get_probability_3(n=10, s=5):
    c = max(0, s - 2 * max(s-n, 1) + s % 2)  # Number of combinations with sum s
    return Fraction(c, n * (n - 1))


if __name__ == "__main__":
    n, s = 10, 5
    print(get_probability_1(n, s))
    print(get_probability_2(n, s))
    print(get_probability_3(n, s))