Combinatorics: counting without listing

A password policy allows 8 characters drawn from 62 symbols: 26 lower-case letters, 26 upper-case and 10 digits. How many passwords is that? Listing them is out of the question. Counting them takes one line: 62⁸ = 218,340,105,584,896, about 2.2 × 10¹⁴.

Whether a brute-force attack is hopeless or a weekend's work depends on that number and the attacker's guess rate. Counting arguments decide the size of a key space, the probability of a hash collision and the running time of an algorithm that tries every arrangement.