In a numerical coding pattern, expressions of the form a θ b δ c are evaluated using the same hidden rule. It is given that 8 θ 12 δ 6 = 60 and 13 θ 15 δ 11 = 74. Using the same rule, what is the value of 18 θ 21 δ 15 ?

Introduction / Context:
This question involves a coded operation that takes three numbers at a time. The symbols θ and δ together represent a fixed numerical rule which combines the three numbers into a single result. We are given two examples of this pattern and must identify the underlying formula so that we can apply it to a third triple of numbers. Recognising how products and squares might be involved is the central reasoning skill tested here.

Given Data / Assumptions:

• 8 θ 12 δ 6 = 60.
• 13 θ 15 δ 11 = 74.
• The symbols θ and δ are part of a single fixed pattern applied to the three numbers.
• We must find 18 θ 21 δ 15 under the same rule.
• Only standard arithmetic operations such as addition, subtraction and multiplication are assumed to be used inside the pattern.

Concept / Approach:
Because we have three numbers and a relatively small result, a good starting point is to try simple formulas built from multiplication and subtraction. In the first example, 8 * 12 is 96 and 6^2 is 36. The difference 96 - 36 equals 60, which matches the result perfectly. For the second example, 13 * 15 is 195 and 11^2 is 121. The difference 195 - 121 equals 74, which again matches the given result. This strongly suggests that the coded pattern is: a θ b δ c = (a * b) - (c^2). Once this structure is identified and verified, we can safely use it for the third triple (18, 21, 15).

Step-by-Step Solution:
Step 1: From the first example, compute 8 * 12 = 96 and 6^2 = 36, then 96 - 36 = 60, which matches 8 θ 12 δ 6. Step 2: From the second example, compute 13 * 15 = 195 and 11^2 = 121, then 195 - 121 = 74, which matches 13 θ 15 δ 11. Step 3: Conclude that the pattern is a θ b δ c = (a * b) - (c^2). Step 4: For 18 θ 21 δ 15, compute the product of the first two numbers: 18 * 21. Step 5: 18 * 21 = 378. Step 6: Compute the square of the third number: 15^2 = 225. Step 7: Subtract: 378 - 225 = 153. Step 8: Therefore, 18 θ 21 δ 15 = 153.

Verification / Alternative check:
The same formula works perfectly for both given examples without any adjustment or extra terms, which is strong evidence that it is the intended rule. If we tried alternative expressions such as (a + b) * c or a * c + b, they would not produce both 60 and 74 from the given data. The use of a product minus a square is neat and matches typical exam patterns. Recomputing 378 - 225 carefully confirms the final result 153, and no other simple formula fits all the information so cleanly.

Why Other Options Are Wrong:
Option 161, 139, 147 and 172 do not equal 378 - 225 and would require changing the rule in a way that breaks the first two examples. For instance, adding instead of subtracting the square, or using the square of a different number, will change the value substantially and make it inconsistent with the given equations. These options act as distractors for students who misidentify which number is being squared or who add instead of subtracting.

Common Pitfalls:
A frequent mistake is to assume that θ and δ each correspond to simple operations such as + or × and to try to evaluate the expression directly without noticing the three number structure. Another error is to square the wrong number (for example, squaring b instead of c) or to forget the square entirely. To avoid this, always check that any proposed pattern fits every given example exactly before using it for the final calculation.

Final Answer:
Using the rule (a * b) - (c^2), the value of 18 θ 21 δ 15 is 153.