If you want license plates consisting of four numbers and four letters in any possible arrangement, then you must first choose $4$ of the $8$ spots for your numbers (or symmetrically, your letters). Then there are $10^4$ choices for your numbers and $26^4$ choices for your letters. This means that in total you have
distinct license plates.
If you make the restriction that the letters come before the numbers (or the numbers before the letters) then you must choose the first four spots for your letters and the last four spots for your numbers. This simply removes the binomial coefficient in the above, leaving $26^4\cdot 10^4$ distinct license plates.