how many passwords?

password_combination.py

import itertools

"""
Consider we have n sets, each contains n_i elements, sets are disjoint with each other

Q: How many distinct k arrangements are there if we want each of
   the n sets to have at least one element in the arrangement ?

"""

def get_count(ns, k):
    n = len(ns)
    char_n = sum(ns)

    result = 0
    for m in range(n, 0, -1):
        sign = (-1) ** (n - m)
        for sets in itertools.combinations(ns, m):
            result += sign * sum(sets) ** k

    return result


def get_count_in_range(ns, k_min, k_max):
    return sum(get_count(ns, k) for k in [k_min, k_max +1])


if __name__ == '__main__':
    # lowercase, uppercase, numbers, special characters
    ns = [26, 26, 10, 33]
    k = 8
    x = get_count()
    print("Number of possible passwords in %s digits: %s" % (k, x))