Question

A computer password contains six characters but does not permit repetition of any characters (either letters or numbers). If uppercase and lowercase letters are used, how many possible passwords are there?

Answer 1

`(62!)/(56!)`

Explanation:

We can either look at all 62 characters (2 * 26 in the alphabet and 10 digits), choose 6 of them, and add the fact that we care about their order - so multiply by 6! and we get
`\binom{62}{6}\cdot 6! = (62!)/(6!\cdot 56!) \cdot 6! = (62!)/(56!)`

Or, we can look atr all the permutations of all the characters and take into account that we DON'T care about the order of the 56 after the first six. Getting
`(62!)/(56!)` immediately.