The question presents a number pattern arranged in three groups, each consisting of three numbers followed by a result.

• Group 1: $2, 9, 5 \rightarrow 24$
• Group 2: $3, 8, ? \rightarrow 504$
• Group 3: $4, 7, 6 \rightarrow 30$

We need to find the missing number (?) in Group 2.

Finding Group 3 Pattern

Let's analyze Group 3: Numbers are $4, 7, 6$ and the result is $30$. Let the numbers be $A=4, B=7, C=6$.

Consider the calculation $(A \times C) + B - 1$: $ (4 \times 6) + 7 - 1 = 24 + 7 - 1 = 31 - 1 = 30 $. This pattern matches the result for Group 3.

Finding Group 1 Pattern

Let's analyze Group 1: Numbers are $2, 9, 5$ and the result is $24$. Let the numbers be $A=2, B=9, C=5$.

Consider the calculation $(A \times B) + C + 1$: $ (2 \times 9) + 5 + 1 = 18 + 5 + 1 = 23 + 1 = 24 $. This pattern matches the result for Group 1.

Finding Group 2 Pattern and Missing Number

Now let's analyze Group 2: Numbers are $3, 8, ?$ and the result is $504$. Let the numbers be $A=3, B=8$, and the missing number be $C=?$. The result $504$ is significantly larger than the other results, suggesting a different calculation pattern.

Let's consider the pattern $(B-C) \times B \times (B+C)$. This pattern uses the middle number $B$ prominently.

For Group 2: $A=3, B=8, C=?$. The result is $504$. Applying the pattern:

$ (8 - C) \times 8 \times (8 + C) = 504 $

Divide both sides by 8:

$ (8 - C) \times (8 + C) = \frac{504}{8} $

$ (8 - C) \times (8 + C) = 63 $

Using the difference of squares formula ($ (x-y)(x+y) = x^2 - y^2 $):

$ 8^2 - C^2 = 63 $

$ 64 - C^2 = 63 $

$ C^2 = 64 - 63 $

$ C^2 = 1 $

Since the numbers in the pattern are generally positive, we take the positive square root:

$ C = 1 $

Thus, the missing number is $1$. The pattern for Group 2 is $(B-C) \times B \times (B+C)$.

Final Answer Derivation

The missing number is determined using the specific pattern identified for the group yielding the result $504$, which is $(B-C) \times B \times (B+C)$. With $B=8$ and $C$ as the missing number, the calculation leads to $C=1$.