Notice that the number of arrangements where B1​ and B2​ are together plus the number of arrangements where they are not is

which is the total number of arrangements.

This property is called complementarity, and is very useful for solving counting questions.

For example, to find the number of 3 digit codes containing at least one 7, you could either

1. Count the number of codes with one 7, two 7s and 3 7s (very painful)

OR

2. Count the number of codes with no 7s, and subtract this number from the total number of codes (very easy)

Option 2 looks like:

So the number of codes with at least one 7 is