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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?

bodygued
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1 Answers1

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

ncmathsadist
  • 49,383
  • Thank you very much! And I did not even notice those two are basically the same so thanks for pointing that out! – bodygued Oct 16 '14 at 01:09