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Mathematics Question on permutations and combinations

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

A

5

B

10

C

8

D

7

Answer

5

Explanation

Solution

1st1^{st} solution : n+1C3−nC3=10^{n+1}C_{3} -^{n}C_{3} = 10
⇒(n+1)n(n−1)6−n(n−1)(n−2)6=10\Rightarrow \frac{\left(n+1\right)n\left(n-1\right)}{6} -\frac{n\left(n-1\right)\left(n-2\right)}{6}=10
⇒3n(n−1)=60⇒n(n−1)=20\Rightarrow 3n\left(n-1\right)=60 \Rightarrow n\left(n-1\right)=20
⇒n2−n−20=0\Rightarrow n^{2} -n -20 =0
⇒(n−5)(n+4)=0   ∴ n=5\Rightarrow \left(n-5\right)\left(n+4\right)= 0 \:\:\: \therefore \:n=5
2nd2^{nd} solution : n+1C3− nC3=10^{n+1}C_{3} - \,^{n}C_{3} =10
⇒ nC2=10⇒n(n−1)2=10\Rightarrow \,^{n}C_{2} =10 \Rightarrow \frac{n\left(n-1\right)}{2}=10
⇒n2−n−20=0. ∴ n=5\Rightarrow n^{2} -n -20 =0.\, \therefore \: n=5
Here we have used nCr+ nCr+1= n+1Cr+1^{n}C_{r}+\,^{n}C_{r+1}=\,^{n+1}C_{r+1}