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:

Step-by-step solution

The number of triangles formed by joining the vertices of an n -sided regular polygon is given by p_n = ^ n C₃ . Given that p_ n+1 - p_n = 66 . ^ n+1 C₃ - ^ n C₃ = 66 Using the property ^ n+1 C_ r - ^ n C_ r = ^ n C_ r-1 , we get: ^ n C₂ = 66 n(n-1) 2 = 66 n(n-1) = 132 n^2 - n - 132 = 0 (n-12)(n+11) = 0 Since n must be a positive integer, n = 12 . The prime factorization of 12 is 2^2 3 . The distinct prime divisors of 12 are 2 and 3 . The sum of all distinct prime divisors of n is 2 + 3 = 5 . Answer: 5