Arrangements with repetition calculator

How many four-digit PINs are there? Each of the four places can take any of the ten digits, and digits may repeat, so 0000 and 7737 both count. That is exactly what arrangements with repetition count: how many ordered k-tuples can be made from n items when items may repeat and the order matters. If items may not repeat, it is an ordinary arrangement, also called a k-permutation.

Enter how many items you are choosing from and how long the string is.

How it works

The number of arrangements of length k from n items, with repetition, is

$$n^k$$

The first place of a PIN can be any of the ten digits. For each of those, the second place can again be any of the ten, which gives 10 · 10 = 100 possibilities. Multiplying in the third and fourth places as well:

$$10\cdot 10\cdot 10\cdot 10={10}^4=\mathrm{10,000}$$

So there are ten thousand four-digit PINs. A thief trying one a second would get through them all in 10,000 seconds, or about 2.78 hours. That is why the bank blocks your card after three wrong attempts.

With letters the numbers grow much faster. A password made of eight lower-case letters has 26⁸ possibilities, which is over 208 billion. You can find out more about what makes a password secure in the article on password security.