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

Step-by-step solution

A polygon with (n+3) sides has (n+3) vertices. A triangle is formed by selecting any 3 vertices out of the total vertices. The number of ways to select 3 vertices from (n+3) vertices is given by the combination formula ^ n+3 C₃ . Given that the number of triangles is 220 , we have the equation: ^ n+3 C₃ = 220 Expanding the combination formula: (n+3)(n+2)(n+1) 3 2 1 = 220 (n+3)(n+2)(n+1) = 220 6 (n+3)(n+2)(n+1) = 1320 We look for three consecutive integers whose product is 1320 . Testing values: 10 11 12 = 1320 Comp