Explanation:

Let’s break it down clearly.

- The pattern for each triplet seems to be: first number × third number × first number = second number.

- Let's check this on the examples you gave and the options.

Given: (2, 114, 19)

- 2 × 19 × 2 = 76. That doesn't match 114.

- Is it addition? 2 + 19 = 21, not 114.

- Is it exponentiation? 2^19 = a huge number. Not matching.

- How about 2 × 19 + 2 = 40. Not 114.

- 2 × (19 + 2) = 42.

So, doesn’t match with simple operators.

Second given: (12, 216, 6)

- 12 × 6 × 12 = 864.

- 12 × 6 = 72.

- 12 × (6 + 12) = 216.

- Hold on, this fits: 12 × (6 + 12) = 12 × 18 = 216.

So maybe the rule is: first number × (third number + first number) = second number.

Let’s apply this rule to all options:

Option 1: (5, 20, 2)

- 5 × (2 + 5) = 5 × 7 = 35, not 20.

Option 2: (6, 144, 8)

- 6 × (8 + 6) = 6 × 14 = 84.

Option 3: (11, 116, 4)

- 11 × (4 + 11) = 11 × 15 = 165, not 116.

Option 4: (4, 128, 7)

- 4 × (7 + 4) = 4 × 11 = 44, not 128.

None of the options give a perfect fit with the pattern from (12, 216, 6).

Let’s try the second possible pattern: first number × third number × some constant?

Trying a division maybe:

- For (2, 114, 19): 114 ÷ 2 = 57. 57 – 19 = 38.

- For (12, 216, 6): 216 ÷ 12 = 18. 18 – 6 = 12.

No clear pattern here.

Another angle: power? (But using whole numbers only.)

Let's check squares or cubes:

- For (12, 216, 6): 6^3 = 216.

So maybe, the second number is the cube of the third.

Check (2, 114, 19): 19^3 = 6859, which is far from 114. So that's out.

What about second number as a square or multiple of third number?

Option 2: (6, 144, 8)

- 8^2 = 64, not 144.

- 6^2 = 36.

- 12^2 = 144, but that’s not an option.

Try other options:

Option 3: (11, 116, 4)

- 4^3 = 64.

- 11^2 = 121.

Option 4: (4, 128, 7)

- 4 x 32 = 128 (no, 7).

Option 1: (5, 20, 2)

- 2^3 = 8.

None of these clearly matches.

Alright, let's reverse the logic: see if original statements have any other pattern.

Let's look at the difference:

- (2, 114, 19): 114 – 2 = 112. 112 ÷ 19 = 5.89.

- (12, 216, 6): 216 – 12 = 204. 204 ÷ 6 = 34.

Not a clean pattern.

It seems like neither product, nor sum, nor cube, nor square of whole numbers using allowed operations yields the given answers.

Given the attempted patterns, none of the options actually fits the pattern established by (12, 216, 6).

Your choice: Option 2 (6, 144, 8)

- 8 × 18 = 144, but we’d need the first number to be 18, not 6, which doesn’t match any rule established.

So, here’s the answer:

- There’s no solid, obvious pattern linking the given triplets to the options, with only allowed operations and no digit splits.

- If we must select, none of these options reproduces the pattern for (12, 216, 6) within the constraints.

- If a “nearest” pattern must be chosen (say, second is the product of numbers, or a square/cube), NONE fits exactly.

In summary: If following the only detectable pattern (first × (third + first) = second), none of the options fits. So none is correct as per the rule.