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

Question

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 (A) 7 (B) 5 (C) 10 (D) 8

Tardigrade Admin · Accepted Answer

Given , Tn= nC3 ⇒ Tn+1 = n+1C3 ∴ Tn+1-Tn = n+1C3 - nC3 =10 [given] ⇒ nC2 + n C3 - nC3 =10 [∵ nCr + nCr+1= n+1Cr+1] ⇒ nC2 =10 ⇒ n=5

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

7

5

10

8

Given , $T_{n} =^{n} C_{3} \Rightarrow T_{n + 1} =^{n + 1} C_{3}$
$∴$ $T_{n + 1} - T_{n} =^{n + 1} C_{3} -^{n} C_{3} = 10$ [given]
$\Rightarrow^{n} C_{2} +^{n} C_{3} -^{n} C_{3} = 10 [∵^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}]$
$\Rightarrow^{n} C_{2} = 10$
$\Rightarrow n = 5$

Q. 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

7

5

10

8

Given , Tn​=nC3​⇒Tn+1​=n+1C3​ ∴ Tn+1​−Tn​=n+1C3​−nC3​=10 [given] ⇒nC2​+nC3​−nC3​=10[∵nCr​+nCr+1​=n+1Cr+1​] ⇒nC2​=10 ⇒n=5

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

Given , $T_{n} =^{n} C_{3} \Rightarrow T_{n + 1} =^{n + 1} C_{3}$
$∴$ $T_{n + 1} - T_{n} =^{n + 1} C_{3} -^{n} C_{3} = 10$ [given]
$\Rightarrow^{n} C_{2} +^{n} C_{3} -^{n} C_{3} = 10 [∵^{n} C_{r} +^{n} C_{r + 1} =^{n + 1} C_{r + 1}]$
$\Rightarrow^{n} C_{2} = 10$
$\Rightarrow n = 5$