Key Concepts and Formulas

• Combinations: The number of ways to choose $k$ items from a set of $n$ distinct items, where the order of selection does not matter, is given by the combination formula: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$
• Number of Triangles from Vertices of a Polygon: The number of triangles that can be formed by joining the vertices of an $n$-sided polygon is $\binom{n}{3}$.
• Combinatorial Identity (Pascal's Identity): $\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$, which can be rearranged to $\binom{n}{k} - \binom{n-1}{k} = \binom{n-1}{k-1}$.

Step-by-Step Solution

Step 1: Define the number of triangles $T_n$

The number of triangles that can be formed by joining the vertices of an $n$-sided polygon is given by choosing 3 vertices out of $n$, which can be represented as: $T_n = \binom{n}{3}$

Step 2: Express $T_{n+1}$

Similarly, the number of triangles that can be formed from an $(n+1)$-sided polygon is: $T_{n+1} = \binom{n+1}{3}$

Step 3: Set up the equation using the given condition

We are given that $T_{n+1} - T_n = 10$. Substituting the expressions for $T_{n+1}$ and $T_n$, we get: $\binom{n+1}{3} - \binom{n}{3} = 10$

Step 4: Apply Pascal's Identity

Using the combinatorial identity $\binom{n}{k} - \binom{n-1}{k} = \binom{n-1}{k-1}$, we can rewrite the left side of the equation. Specifically, we can write $\binom{n+1}{3} - \binom{n}{3} = \binom{n}{2}$ Thus, our equation becomes: $\binom{n}{2} = 10$

Step 5: Expand the combination and solve for $n$

Expanding the combination $\binom{n}{2}$, we have: $\binom{n}{2} = \frac{n!}{2!(n-2)!} = \frac{n(n-1)}{2}$ Substituting this into our equation: $\frac{n(n-1)}{2} = 10$ Multiplying both sides by 2: $n(n-1) = 20$ Expanding and rearranging into a quadratic equation: $n^2 - n - 20 = 0$ Factoring the quadratic: $(n-5)(n+4) = 0$ This gives us two possible solutions for $n$: $n = 5 \quad \text{or} \quad n = -4$

Step 6: Validate the solution

Since $n$ represents the number of sides of a polygon, it must be a positive integer. Therefore, $n = -4$ is not a valid solution. Also, we need $n \ge 3$ for the combination $\binom{n}{3}$ to be defined. The solution $n=5$ satisfies this condition.

Therefore, the value of $n$ is 5.

Common Mistakes & Tips

• Incorrectly applying the combination formula: Ensure you understand the combination formula and apply it correctly when calculating the number of triangles.
• Forgetting to validate the solution: Always check if the solution obtained makes sense in the context of the problem. The number of sides of a polygon must be a positive integer greater than or equal to 3.
• Not recognizing Pascal's Identity: Pascal's Identity simplifies the problem significantly. Learn to recognize situations where it can be applied.

Summary

We determined the number of triangles that can be formed from an $n$-sided polygon as $\binom{n}{3}$. We then used the given condition $T_{n+1} - T_n = 10$ and Pascal's Identity to simplify the equation to $\binom{n}{2} = 10$. Solving this equation resulted in $n=5$ or $n=-4$. Since $n$ must be a positive integer, we validated that $n=5$ is the correct solution.

Final Answer

The final answer is \boxed{5}, which corresponds to option (B).