Step 1: Read the pattern from the shapes given. Each polygon has a certain number of sides, and a number is written inside it. A triangle has \(3\) sides and shows \(6\). A rectangle has \(4\) sides and shows \(24\). A pentagon has \(5\) sides and shows \(120\). We need to find the rule linking the side count \(n\) to the number shown, then apply it to the hexagon, which has \(6\) sides.
Step 2: Test the factorial rule. Check whether the number shown equals \(n!\) (n factorial), where \(n\) is the number of sides. For the triangle, \(n=3\): \(3! = 3 \times 2 \times 1 = 6\). This matches the \(6\) shown. For the rectangle, \(n=4\): \(4! = 4 \times 3 \times 2 \times 1 = 24\). This matches the \(24\) shown. For the pentagon, \(n=5\): \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). This matches the \(120\) shown. The rule holds for all three given shapes, so \(n!\) is the pattern connecting sides to the number inside.
Step 3: Apply the rule to the hexagon. A hexagon has \(n=6\) sides, so \[ X = 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \]
Step 4: Check why the other options are wrong. Option (B) \(596\) and option (D) \(240\) do not come from any factorial or simple multiple of the pattern; they are plain distractors that do not fit \(n!\). Option (C) \(24\) is the value already used for the rectangle (\(n=4\)), not the hexagon, so picking it means forgetting to move from \(n=4\) to \(n=6\).
Final Answer: The value of \(X\) is \(720\), so option (A) is correct. \[ \boxed{X = 720} \]
