Select the option that is related to the third letter-number cluster in the same way as the first letter-number cluster is related to the second letter-number cluster, and the fifth letter-number cluster is related to the sixth letter-number cluster. (Note: Mathematical operations must be applied to the complete numbers themselves, not to their individual digits.) JK180 : PQ360 :: CD210 : ? :: ST140 : YZ280

The correct answer is IJ420.

To solve this letter-number cluster analogy, we need to analyze the relationship between both the letters and the numbers in the given pairs: JK180 : PQ360 and ST140 : YZ280.

Step 1: Analyze the Pattern in the Letters
Let's examine the alphabetical positions of the letters (where A = 1, B = 2, C = 3, ..., Z = 26):

For the first pair (JK to PQ):

$J\left(10\right)+6=P\left(16\right)$

$K\left(11\right)+6=Q\left(17\right)$

For the third pair (ST to YZ):

$S\left(19\right)+6=Y\left(25\right)$

$T\left(20\right)+6=Z\left(26\right)$

Thus, each letter is shifted forward by 6 positions (+6 pattern).

Now, apply this +6 letter pattern to CD:

$C\left(3\right)+6=I\left(9\right)$

$D\left(4\right)+6=J\left(10\right)$

So, the letter part of the missing cluster is IJ.

Step 2: Analyze the Pattern in the Numbers
Next, look at the relationship between the numbers in each pair:
In the first pair (180 to 360):

$180\times 2=360$

In the third pair (140 to 280):

$140\times 2=280$

Thus, the number is multiplied by 2.

Now, apply this number pattern to 210:

$210\times 2=420$

So, the numerical part of the missing cluster is 420.

Step 3: Combine Letters and Numbers
Combining the resulting letters (IJ) and number (420), we get IJ420.

Hence, the correct option is IJ420.