Question

Based on the relationship between each polygon and the number inside it, the value of \(X\) is ______

• A. 720
• B. 596
• C. 24
• D. 240

Hint:

Compare the number of sides of each polygon with the number written inside it, and test the factorial rule.

Answer 1

The Correct Option is A

Look at how the number inside each shape grows from one shape to the next, using ratios between consecutive shapes instead of jumping straight to a factorial formula.

1. Triangle to rectangle: the triangle (3 sides) shows $6$ and the rectangle (4 sides) shows $24$. The ratio is $24/6 = 4$, which is the side count of the rectangle.
2. Rectangle to pentagon: the rectangle (4 sides) shows $24$ and the pentagon (5 sides) shows $120$. The ratio is $120/24 = 5$, which is the side count of the pentagon.
3. Pentagon to hexagon: following the same growth rule, going from the pentagon (5 sides) to the hexagon (6 sides) should multiply the value by $6$, the side count of the hexagon.

So $X = 120 \times 6 = 720$. This lines up with building each term as $n!$ directly, since $6! = 720$ too.

Let's summarize:

• Each shape's number equals the previous shape's number times its own side count, which is exactly how a factorial builds up step by step.
• Starting the chain at the triangle's $6 = 3!$ and repeatedly multiplying by the next side count reproduces $24$, $120$, and finally $720$ for the hexagon.

The value of $X$ is $720$, matching option (A).