Binomial TheoremMedium
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
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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