Here's a different approach...

Take the task of creating codes and break it into stages.

Stage 1: Determine the GENERIC arrangement of digits and letters.
For example, one possible arrangement is DIGIT-LETTER-LETTER-DIGIT-LETTER
Another possible arrangement is LETTER-DIGIT-DIGIT-LETTER-LETTER

Let L represent a letter
Let D represent a digit
So we want to arrange 3 L's and 2 D's

-----ASIDE------
When we want to arrange a group of items in which some of the items are identical, we can use something called the MISSISSIPPI rule. It goes like this:

If there are n objects where A of them are alike, another B of them are alike, another C of them are alike, and so on, then the total number of possible arrangements = n!/[(A!)(B!)(C!)....]

So, for example, we can calculate the number of arrangements of the letters in MISSISSIPPI as follows:
There are 11 letters in total
There are 4 identical I's
There are 4 identical S's
There are 2 identical P's
So, the total number of possible arrangements = 11!/[(4!)(4!)(2!)]
--------------------------

We want to arrange 3 L's and 2 D's. So:
There are 5 characters in total
There are 3 identical L's
There are 2 identical D's
So, the total number of possible arrangements = 5!/[(3!)(2!)] = 10
So, we can complete stage 1 in 10 ways

Stage 2: Select a letter (X or Y) to replace the 1st L in the arrangement (the arrangement from stage 1)
We can choose X or Y, so we can complete stage 2 in 2 ways

Stage 3: Select a letter (X or Y) to replace the 2nd L in the arrangement (the arrangement from stage 1)
We can choose X or Y, so we can complete stage 3 in 2 ways

Stage 4: Select a letter (X or Y) to replace the 3rd L in the arrangement (the arrangement from stage 1)
We can choose X or Y, so we can complete this stage in 2 ways

Stage 5: Select a digit (1, 2 or 3) to replace the 1st D in the arrangement (the arrangement from stage 1)
We can choose 1, 2 or 3, so we can complete this stage in 3 ways

Stage 6: Select a digit (1, 2 or 3) to replace the 2nd D in the arrangement (the arrangement from stage 1)
We can choose 1, 2 or 3, so we can complete this stage in 3 ways

By the Fundamental Counting Principle (FCP), we can complete all 6 stages (and thus create a code) in (10)(2)(2)(2)(3)(3) ways (= 720 ways)

Answer: D

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

RELATED VIDEOS

Thanks

short and sweet. thanks

will not appear on GRE at this level of difficulty

Hello from the GRE Prep Club BumpBot!

Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.