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nCr Definitions and Properties

Let Tn deno...

Question

Let $T_{n}$ denote the number of triangles which can be formed using the vertices of a regular polygon on n sides. If $T_{n + 1} - T_{n} = 21$ then n equals

A

5

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B

7

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C

6

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D

4

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Solution

The correct option is B $7$
From a polygon any 3 points can form a triangle.
Hence $T_{n} = (\begin{matrix} n 3 \end{matrix}) t h u s T_{n + 1} = (\begin{matrix} n + 1 3 \end{matrix}) A / Q T_{n + 1} - T_{n} = 21 \Rightarrow (\begin{matrix} n + 1 3 \end{matrix}) - (\begin{matrix} n 3 \end{matrix}) = 21 \Rightarrow n = 7$
Hence, option 'B' is correct.

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