Verified Solution • 2025–26 Session moderate

QJEEM13Ma9c4c: JEE Main 2013 Mathematics - Class 11 JEEM13Ma9c4c Solution: Let be the number of all possible triangles formed by joining…

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

Multiple Choice Options

  • A 7
  • B 5
  • C 10
  • D 8

Step-by-Step Verified Solution

Number of possible triangle using n vertices = nC3

∴\therefore Tn = nC3

then Tn + 1 = n + 1C3

Given, Tn+1−Tn{T_{n + 1}} - {T_n} = 10

⇒\Rightarrow n + 1C3 - nC3 = 10

⇒\Rightarrow (n+1)n(n−1)6−n(n−1)(n−2)6{{\left( {n + 1} \right)n\left( {n - 1} \right)} \over 6} - {{n\left( {n - 1} \right)\left( {n - 2} \right)} \over 6} = 10

⇒\Rightarrow 3n(n - 1) = 60

⇒\Rightarrow n(n - 1) = 20

⇒\Rightarrow n2 - n - 20 = 0

⇒\Rightarrow (n - 5)(n + 4) = 0

∴\therefore n = 5

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