Permutations with Repetition Calculator

Every position keeps all n choices, so the count is n multiplied by itself r times. Ten digits on four positions give 10 × 10 × 10 × 10 = 10,000 PINs; two symbols on eight positions give 256, one byte. Forbid repeats and the same question becomes the ordinary permutations calculator.

Of the four counting formulas this is the plainest, and the reason deserves saying out loud: it is the only one with nothing to correct. Each of the r positions is filled independently, and because nothing is consumed, each keeps all n options. The multiplication principle then runs straight through — n choices, times n, times n, r times over — and that is n^r. The other three formulas all end in an adjustment: a product that shrinks because items get used up, or a division that takes back orders counted more than once. Here nothing is used up and every order really is a different result, so no factorial appears anywhere. Ten digits on four positions give 10 × 10 × 10 × 10 = 10,000, which is also the plain reading of a four-digit PIN: the codes 0000 through 9999, ten thousand of them exactly. What is counted is possibilities, not difficulty. Eight printable ASCII characters span 6,634,204,312,890,625 strings, but that figure only describes how strong a password would be if it were drawn uniformly at random from the whole space. People do not choose that way, and an attacker does not work through the space in order. The number is an upper bound on the effort, never a measure of one particular choice.