Question

Let \( p_n \) denote the total number of triangles formed by joining the vertices of an \( n \)-side regular polygon. If \( p_{n+1} - p_n = 66 \), then the sum of all distinct prime divisors of \( n \) is:

• A. 7
• B. 8
• C. 5
• D. 6

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To form a triangle, we need to choose 3 distinct vertices from the \(n\) vertices of the polygon.
The number of ways to do this is given by the combination formula \(\binom{n}{3}\).
Step 2: Key Formula or Approach:
The total number of triangles formed is \(p_n = \binom{n}{3}\).
We are given the mathematical relation:
\[ p_{n+1} - p_n = 66 \] Substituting the combination formula into this relation yields:
\[ \binom{n+1}{3} - \binom{n}{3} = 66 \] Step 3: Detailed Explanation:
Using Pascal's Identity \(\binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k}\), we can rewrite the left side.
\[ \binom{n+1}{3} - \binom{n}{3} = \binom{n}{2} \] So, we have a simplified equation.
\[ \binom{n}{2} = 66 \] Expand the combination formula for further solving.
\[ \frac{n(n-1)}{2} = 66 \implies n(n-1) = 132 \] We need to find two consecutive positive integers whose product is 132.
Since \(11 \times 12 = 132\), we can determine that \(n = 12\).
The prime factorization of \(12\) is \(2^2 \times 3\).
The distinct prime divisors of \(12\) are \(2\) and \(3\).
The sum of these distinct prime divisors is \(2 + 3 = 5\).
Step 4: Final Answer:
The sum of all distinct prime divisors of \(n\) is \(5\).