Permutations and Combinations 2 Question 11
11. 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
(2013 Main)
(a) 7
(b) 5
(c) 10
(d) 8
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Answer:
Correct Answer: 11. (b)
Solution:
- Given, $T _n={ }^{n} C _3 \Rightarrow T _{n+1}={ }^{n+1} C _3$
$$ \begin{array}{lrlr} \therefore & T _{n+1}-T _n & ={ }^{n+1} C _3-{ }^{n} C _3=10 & \text { [given] } \\ \Rightarrow & { }^{n} C _2+{ }^{n} C _3-{ }^{n} C _3 & =10 & {\left[\because{ }^{n} C _r+{ }^{n} C _{r+1}={ }^{n+1} C _{r+1}\right]} \\ \Rightarrow & { }^{n} C _2 & =10 \\ \Rightarrow & n & n & \end{array} $$
