We know that an employee generates a series of five-digit product codes.
_ _ _ _ _
The codes use only the digits 0, 1, 2, 3, and 4 and each of these digits is used exactly once in the code.
Rule 1 - The second digit has a value exactly twice that of the first digit.
Therefore, there are only two possibilities for the first two digits:
(1) 1 and 2 (2) 2 and 4
Rule 2 - The value of the third digit is less than the value of the fifth digit.
These rules are pretty restrictive. As there are only five digits and the placement of two has already been determined by Rule 1, we can quickly go through and outline all possible product codes.
Let's start with those beginning with 1 and 2. We know that the third digit must be less than the fifth digit.
This means that 0 cannot be the fifth digit and 4 cannot be the third digit when 3 is the fifth digit. This gives us three possible product codes.
Code #1: 1 2 3 0 4
Code #2: 1 2 0 3 4
Code #3: 1 2 0 4 3
Now let's figure out those codes beginning with 2 and 4.
Again, 0 cannot be the fifth digit. We also know that 3 cannot be the third digit when 1 is the fifth digit. This, again, gives us three possible product codes.
Code #4: 2 4 1 0 3
Code #5: 2 4 0 1 3
Code #6: 2 4 0 3 1
Now that we have a comprehensive list of all possible product codes, let's address the question stem. We are looking for the answer choice which must be true about any acceptable product code. As we have all the acceptable product codes listed above, we just have to look for one code which contradicts an answer choice.
Answer choice (A) is incorrect because Code #1 has two digits in between 1 and 0.
Answer choice (B) is incorrect because Code #6 has three digits in between 1 and 2.
Answer choice (C) is incorrect because Code #3 has three digits in between 1 and 3.
Answer choice (D) is incorrect because Code #4 has three digits in between 2 and 3.
Answer choice (E) is correct because Codes 4-6 have zero digits in between 2 and 4, Codes 1 and 2 have two digits in between 2 and 4, and Code 3 has one digit in between 2 and 4. Therefore, it must be true that there are at most two digits between 2 and 4 because none of the possible product codes have three digits in between 2 and 4.
Hope this helps! Please let us know if you have any further questions.
