Question

The number of triangles which can be formed by using the vertices of a regular polygon of \( (n + 3) \) sides is 220. Then, \( n \) is equal to:

• A. 8
• B. 9
• C. 10
• D. 11

Hint:

When counting combinations, use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), and solve for the given value.

Answer 1

The Correct Option is B

We are given that the number of triangles that can be formed from \( (n+3) \) sides is 220. Step 1: Use the combination formula The number of triangles that can be formed from \( (n + 3) \) vertices is given by: \[ \binom{n+3}{3} = 220 \] Step 2: Solve for \( n \) We know that: \[ \binom{n+3}{3} = \frac{(n+3)(n+2)(n+1)}{6} \] Set this equal to 220: \[ \frac{(n+3)(n+2)(n+1)}{6} = 220 \] Multiply both sides by 6: \[ (n+3)(n+2)(n+1) = 1320 \] Step 3: Solve the cubic equation Expanding the left side: \[ n^3 + 6n^2 + 11n + 6 = 1320 \] \[ n^3 + 6n^2 + 11n - 1314 = 0 \] By trial, we find \( n = 9 \). Thus, the correct answer is \( n = 9 \).