In a certain region all license plates are composed of three letters followed by three numbers, or three numbers followed by three letters.

There are many ways to fill the license plate in. To demonstrate it, we create a template:

 

_ _ _ _ _ _ = 26 * 26 * 26 * 10 * 10 * 10

 

There are $26^3$ ways to choose the letters and $10^3$ ways to choose the numbers. Thus, the number of license plates are $26^3 \cdot 10^3= 17576000.$

Thanks Pangolin,

You forgot that the letters and numbers can be reversed so you will need to multiply your answer by 2.