Key Concepts and Formulas

• Combinations: The number of ways to choose r items from a set of n items (without regard to order) is given by the binomial coefficient: ${}^nC_r = \frac{n!}{r!(n-r)!}$
• Fundamental Principle of Counting: If there are m ways to do one thing and n ways to do another, then there are m * n* ways to do both.

Step-by-Step Solution

Step 1: Calculate the number of triangles ($\alpha$)

To form a triangle, we need to choose three points. The problem specifies that these points must be from different sides of the rectangle. We have four sides (AB, CD, BC, DA) with 5, 7, 6, and 9 points, respectively. We need to choose one point from each of three different sides. We have four possible combinations of three sides:

• AB, BC, CD: Choosing 1 point from each of these sides gives ${}^5C_1 \cdot {}^6C_1 \cdot {}^7C_1$ triangles.
• AB, BC, DA: Choosing 1 point from each of these sides gives ${}^5C_1 \cdot {}^6C_1 \cdot {}^9C_1$ triangles.
• AB, CD, DA: Choosing 1 point from each of these sides gives ${}^5C_1 \cdot {}^7C_1 \cdot {}^9C_1$ triangles.
• BC, CD, DA: Choosing 1 point from each of these sides gives ${}^6C_1 \cdot {}^7C_1 \cdot {}^9C_1$ triangles.

Therefore, the total number of triangles is the sum of these possibilities:

$\alpha = {}^5C_1 \cdot {}^6C_1 \cdot {}^7C_1 + {}^5C_1 \cdot {}^6C_1 \cdot {}^9C_1 + {}^5C_1 \cdot {}^7C_1 \cdot {}^9C_1 + {}^6C_1 \cdot {}^7C_1 \cdot {}^9C_1$

$\alpha = (5 \cdot 6 \cdot 7) + (5 \cdot 6 \cdot 9) + (5 \cdot 7 \cdot 9) + (6 \cdot 7 \cdot 9)$

$\alpha = 210 + 270 + 315 + 378 = 1173$

Step 2: Calculate the number of quadrilaterals ($\beta$)

To form a quadrilateral, we need to choose four points, one from each of the four sides of the rectangle. The number of ways to do this is:

$\beta = {}^5C_1 \cdot {}^6C_1 \cdot {}^7C_1 \cdot {}^9C_1$

$\beta = 5 \cdot 6 \cdot 7 \cdot 9 = 1890$

Step 3: Calculate $\beta - \alpha$

We are asked to find the value of $\beta - \alpha$. We have calculated $\alpha = 1173$ and $\beta = 1890$. Therefore:

$\beta - \alpha = 1890 - 1173 = 717$

Common Mistakes & Tips

• Carefully consider the problem's constraints: The points must be from different sides. Don't consider combinations where multiple points come from the same side.
• Double-check calculations: Arithmetic errors are easy to make, especially when dealing with multiple multiplications.
• Understand the difference between permutations and combinations. Order matters for permutations, but not for combinations. This problem involves combinations because the order in which we choose the points does not affect the triangle or quadrilateral formed.

Summary

We first calculated the number of triangles ($\alpha$) that can be formed by selecting one point from each of three different sides of the rectangle. Then, we calculated the number of quadrilaterals ($\beta$) that can be formed by selecting one point from each of the four sides. Finally, we subtracted $\alpha$ from $\beta$ to obtain the answer.

Final Answer

The final answer is \boxed{717}, which corresponds to option (A).