Question

How many computer passwords can be

created using 3 digits followed by 4 lower-case
letters if no repeats are allowed.

Possible answers left
132
1881600
258336000

Answer 1

There are 10 choices for a given digit (0-9), but without repeating this means we have 10•9•8 = 720 ways of choosing the numerical part of the password. We can also write this as

`(10!)/((10-3)!) = (10!)/(7!)`

There are 26 choices for a given letter character (a-z), and without repetition we have 26•25•24•23 = 358,800 ways of choosing the letters in the password. Similarly, we can write

`(26!)/((26-4)!) = (26!)/(22!)`

Then the total number of passwords is

`(10!26!)/(7!22!) = \boxed{258,336,000}`