Explanation:

Alright, let’s break this problem down so it actually makes sense.

First, look at the two given sets:

(11, 10, 21)

(14, 6, 160)

What’s the pattern? Let’s investigate:

- Set 1: (11, 10, 21) ? 11 + 10 = 21. Simple addition.

- Set 2: (14, 6, 160) ? 14 × 6 = 84. That doesn’t give 160, so what else? Let’s try (14-6)=8, but that doesn’t help. How about powers? 14^2 - 6^2 = 196 - 36 = 160. Bingo!

So, the relationship seems to be this:

a^2 - b^2 = c

Let’s run through the options and check which one fits:

- (16, 9, 165): 16^2 - 9^2 = 256 - 81 = 175 (not 165)

- (19, 12, 217): 19^2 - 12^2 = 361 - 144 = 217

- (81, 16, 65): 81^2 - 16^2 = 6561 - 256 = 6305 (not 65)

- (27, 18, 9): 27^2 - 18^2 = 729 - 324 = 405 (not 9)

So here’s the point-by-point rundown:

- The set uses the rule a^2 - b^2 = c.

- Only option 2 (19, 12, 217) fits this pattern perfectly.

- The rest don’t align—the math doesn’t check out when you use whole numbers, as per the instructions.

The correct answer is Option 2: (19, 12, 217)