Binomial TheoremHard

Question

Let Tn denotes the number of triangles which can be formed using the vertices of a regular polygon of n sides. If Tn+1 - Tn = 21, then n equals

Options

A.5
B.7
C.6
D.4

Solution

According to given condition Tn = nC3
and       Tn+1 - Tn = 21
⇒       n+1C3 - nC3 = 21
⇒       (n + 1)(n)(n - 1) - n(n - 1)(n - 2) = 21
⇒       [(n + 1) - (n - 2)] = 21
⇒       = 21
⇒       n(n - 1) = 42
⇒       n = 7

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