What is the probability that a 6-digit phone number contains at least one 2? (Repetition of numbers and lead zero are allowed).
Imagine the situation as being a room with 6 phones. Each phone has 10 digits on it, and you have to pick one digit from each phone!
As such there are 10 different ways you can choose 1 from 10. And you have to choose 6 of these. Hence there are $\displaystyle 10^6$ combinations!
Now lets consider the same situation, only now all the 2s on the phones have been removed. So now there are 6 phones, each with only 9 digits, (0,1,3,4,5,6,7,8,9). So you are choosing 1 button from 9, 6 times! Hence $\displaystyle 9^6$.
So we have concluded that there are $\displaystyle 10^6 $ combinations, and $\displaystyle 9^6 $ of them have no 2!
Hence the probability that there is at least one 2 is :
$\displaystyle 10^6 - 9^6 = 468559$