License plate combination California's license plate is made up for a number, followed by 3 letters, and 3 more numbers. If you cannot have the word BOB then how many license plate can be made in total? 
I'm guessing it's $10^4 * 26^3 - 10^4$ because the word BOB is disallowed so any combinations that contain that word is not allowed. For example: 1BOB234 is not allowed just like 6BOB986 is also not allowed. So there are a total of 10^4 combinations with the word BOB, that is why I subtracted it from the total numbers of license plates can be made.
However, I think it could also be $10^4 * (26^3 - 1)$ because if I take out one combination with BOB, then I would never have it.
If I did $10^4 * 25^3$ (take out the 3 letters) then I would have forbidden the three letters, hence making the combinations of OBB or BBO not possible while they are allowed.
So how would I approach this problem?

Your initial guess is correct. Also, note that $10^4(26^3-1) = 10^4\cdot 26^3 - 10^4.$
