# How many zip codes are as large as 6000-0000, are even numbers, and have a 7 as their third digit?

When they were first introduced, postal zip codes were five digit numbers, theoretically ranging from $00000$ to $99999$. (In reality, the lowest zip code was $00601$ for San Juan, Puerto Rico; the highest was $99950$ for Ketchikan, Alaska.) An additional four digit have been added, so each zip code is now a nine digit number. How many zip codes are at least as large as 60000-0000, are even numbers, and have a $7$ as their third digit?

Attempt: $A1*A2*A3*...*A9$ be a zip code. With $A3 = 7$. I don't know how to continue.

To be at least as large as 60000-0000 the first place must be one of 6,7,8,9.

To be an even number, the last place must be one of 0,2,4,6,8.

For the third place to be 7 there's only one choice for the third place. 7

The remaining six places can each be any of the ten digits, 0,1,2,3,4,5,6,7,8,9.

Each place is selected independently (digits can be repeated).

Count the ways to do this.

• So the total N = 4*10*1*10*10*10*10*10*5 ? – user3459 Sep 30 '14 at 0:44