## Combinations & Sequences

Lets start with the mathematics behind the length and number of symbols in a sequence. At its most basic a password can be thought of as a sequence of symbols or characters. If a password were composed of one symbol from a list of 10 (0-9), it is trivial to guess. If it is chosen from a list of 1,112,064 characters (UTF-8), it is harder to guess.

The number of possible sequences for 0-9 can be counted using the simple base 10 number system. One place has 10 possible symbols: 0-9. As you add places, you are increasing the number of possibilities by 10 times. Counting the number of possibilities is easy; with five places, there are 100,000 possible combinations (0-99,999) or ${\sf 10^5}$.

Generalizing this, it can be represented as $b^n$ with $b$ the base size (or cardinality) of the set of symbols to choose from (0-9 in this example) and $n$ the total length of the symbols sequenced together (5 in the case of a number like 99,999). Assuming symbols are allowed to repeat, any password that is composed of lower case letters from the english language (a-z, 26 characters) and upper case letters (A-Z, 26 characters), and the numbers (0-9, 10 characters), can be represented as having ${\sf (26+26+10)^n }$ possible sequences or ${\sf 62^n}$.

If a password is chosen from these 62 symbols and is 10 characters long, then the total number of possible passwords is ${\sf 62^{10}}$ or 839,299,365,868,340,224 different possible combinations. If more symbols from standard US keyboards are included in the list, such as +=!@#\$%^&*()_-\"|'?/{}[]`~,.<> an additional 31 symbols are available for a total of 92 and a password with a length of 10, or ${\sf 92^{10}}$, has 43,438,845,422,363,213,824 possible combinations (almost 52 times more than ${\sf 62^{10}}$). As more characters are added to the length of the password, the possibilities increase faster for the set composed of 92 characters compared the set of 62 symbols. For a password of length 15, ${\sf 92^{15}}$ has 372 times more possibilities than ${\sf 62^{15}}$.

This is why some software and services require you to choose passwords that include symbols that are drawn from punctuation; every extra possible symbols helps.