This question involves a specific coding or transformation pattern applied to words. We need to identify this pattern using the given examples ('CERTAIN' -> 'ECTRCGP' and 'DIGITAL' -> 'FGIGVYN') and then apply it to the word 'FACTORY' to find its corresponding code.

Decoding the 'CERTAIN' Transformation

Let's analyze the transformation of 'CERTAIN' to 'ECTRCGP' by looking at the position of each letter in the alphabet.

• C is the 3rd letter, E is the 5th letter. (3 + 2 = 5)
• E is the 5th letter, C is the 3rd letter. (5 - 2 = 3)
• R is the 18th letter, T is the 20th letter. (18 + 2 = 20)
• T is the 20th letter, R is the 18th letter. (20 - 2 = 18)
• A is the 1st letter, C is the 3rd letter. (1 + 2 = 3)
• I is the 9th letter, G is the 7th letter. (9 - 2 = 7)
• N is the 14th letter, P is the 16th letter. (14 + 2 = 16)

The pattern observed is a sequence of operations: +2, -2, +2, -2, +2, -2, +2 applied to the alphabetical positions of the letters.

Verifying the Pattern with 'DIGITAL'

Let's check if this same pattern (+2, -2, +2, -2, +2, -2, +2) works for the second example, 'DIGITAL' -> 'FGIGVYN'.

• D (4) + 2 = 6 (F)
• I (9) - 2 = 7 (G)
• G (7) + 2 = 9 (I)
• I (9) - 2 = 7 (G)
• T (20) + 2 = 22 (V)
• A (1) - 2 = -1. In alphabet coding, -1 wraps around to the end, so it's equivalent to 25 (Y). (1 - 2 = 25)
• L (12) + 2 = 14 (N)

The pattern holds true for 'DIGITAL' as well.

Applying the Coding Pattern to 'FACTORY'

Now, we apply the established pattern (+2, -2, +2, -2, +2, -2, +2) to the word 'FACTORY'.

• F (6) + 2 = 8 => H
• A (1) - 2 = -1 => Y (wrapping around from A)
• C (3) + 2 = 5 => E
• T (20) - 2 = 18 => R
• O (15) + 2 = 17 => Q
• R (18) - 2 = 16 => P
• Y (25) + 2 = 27 => A (wrapping around from Y)

The coded word for 'FACTORY' is 'HYERQPA'.

Final Result and Correct Option

Based on the analysis, the coded form of 'FACTORY' is 'HYERQPA'. This corresponds to the first option provided.