Let Tn be the number of all possible triangles formed by joining vertices of an n-sided regular polygon. If Tn+1 - Tn = 10, then the value of n is?

we know that a triangle needs 3 vertices to be formed so out of n vertices 3 should be chosen = nC3
According to the given condition, we can write that
n+1C3 - nC3 = 10
=> nC2 = 10
=> n(n-1)/2 = 10
=> n(n-1) = 20
=> n² - n - 20 = 0
=> (n-5)(n+4) = 0
=> n = 5 or n = -4
Since n represents the number of sides of a polygon, it must be a positive integer. Therefore, n = 5.