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License Plates

Q:  How many license plates can be made using either three digits followed by three letters or three letters followed by three digits?

A: Back to our trusty “slot method” as shown in many of our other counting problems…

Three digits followed by three letters — that means 6 total slots:

_ _ _ _ _ _

How many choices for the first slot? Well, it has to be a single digit.. Either 0, 1, 2, 3, …, 9.  There are 10 choices (remember to count 0 through 9).

10 _ _ _ _ _

How many choices for the second & third slots? They both have to be digits, so we know there are 10 choices each:

10 10 10 _ _ _

Now, the remaining three slots have to be letters.  There are 26 letters, so there are 26 choices for each of the remaining slots:

10 10 10 26 26 26

So, there are 10*10*10*26*26*26 possible license plates possible ( that have 3 digits followed by 3 letters).

Now, you have to do this for 3 letters followed by 3 digits, to get: 26*26*26*10*10*10 and add the two answers together, which gives:  2*(10*10*10*26*26*26) as a final answer.

My “quiz” is this…

(1) What would the answer look like if the question were three letters followed by three numbers?

(2) What would the answer look like if it was two letters followed by four numbers?

Leave your comments to answer the “quiz”!

5 thoughts on “License Plates

  1. 1) Use the slot method
    6 slot _ _ _ _ _ _

    put the letters first three slots

    26 26 26 _ _ _

    next add numbers

    26 26 26 10 10 10

    ans: 26*26*26*10*10*10 possible license plates

    2) two letters followed by four numbers

    _ _ _ _ _ _

    first the two letters

    26 26 _ _ _ _

    then followed by 4 numbers

    26 26 10 10 10 10

    ans: 26*26*10*10*10*10 possible license plates

  2. Nice work on the “quiz”. The “slot method” makes things much easier, huh?

  3. haha sure does! thanks a million!

  4. Hi, I found your blog just browsing through some of the pages of Google search,

    For the question :”How many license plates can be made using either three digits followed by three letters or three letters followed by three digits?”

    I think the answer should be 2*(26^3*10^3) ? I don’t really understand why are you avoiding the case of ” three letters or three letters followed by three digits”

    Regards,

  5. Debanjan, you are exactly right — I overlooked that (probably because I was in a hurry) — I will fix it right now.

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