No math required. Almost. When we have multiple options for a series of decisions, we can multiply the number of options for each decision. In this case, the first two decisions are which letter to choose, and the last three decisions are which digit to choose. Since there are 26 letters, and 10 digits, and repetition is allowed, we can multiply as follows:

26x26x10x10x10 = ?

Advanced math students should know what 26 squared is, but we don't need to actually multiply these numbers to solve this problem. Whatever the number is, it must end in 3 zeros. Notice only one answer choice does so: E.
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The number of possible license plates is 26 x 26 x 10 x 10 x 10 = 676,000.

Answer: E
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Meanwhile, in State X, in the Governor's office, the following is unfolding....

GOVERNOR: How are things going with the new license plate format?
HEAD OF SUCH THINGS: It’s going really well. We’ll be voting on a proposed format at today’s meeting.
GOVERNOR: Fantastic! How many different license plates are possible with the proposed format?
HEAD OF SUCH THINGS: 23,400
GOVERNOR: Who said what now?

So, unless we’re talking about some 51st state with only a handful of cars in it, we can safely eliminate A, B, C (just kidding)


All kidding aside, we can take the task of creating license plates and break it into stages.

Stage 1: Select the 1st character.
Since the 1st character must be a letter, we can complete stage 1 in 26 ways

Stage 2: Select the 2nd character.
Since this character must be a letter, and since we can REPEAT letters, we can complete this stage in 26 ways

Stage 3: Select the 3rd character.
Since this character must be a digit (0,1,2,3,4,5,6,7,8 or 9), we can complete this stage in 10 ways

Stage 4: Select the 4th character.
Since this character must be a digit, and since we can REPEAT digits, we can complete this stage in 10 ways

Stage 5: Select the 5th character.
Since this character must be a digit, we can complete this stage in 10 ways

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus create a license place) in (26)(26)(10)(10)(10) ways (= 676,000 ways)

Answer: E

Note: the FCP can be used to solve the MAJORITY of counting questions on the GRE. So, be sure to learn it.

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