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nCr Definitions and Properties

Tm denotes th...

Question

$T_{m}$ denotes the number of triangles that can be formed with the vertices of regular polygon of $m$ sides. If $T_{m + 1} - T_{m} = 15$ , then $m =$

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Solution

The correct option is C $6$
Given : $T_{m + 1} - T_{m} = 15$
As $T_{m}$ denotes the number of triangles formed
$\Rightarrow^{(m + 1)} C_{3} -^{m} C_{3} = 15$
$\Rightarrow \frac{(m + 1)!}{3! \times (m - 2)!} - \frac{m!}{3! \times (m - 3)!} = 15$
$\Rightarrow m \times (m - 1) = 6 \times 5$
$\Rightarrow m^{2} - m - 30 = 0$
$\Rightarrow (m - 6) (m + 5) = 0$
$\Rightarrow m = 6$ , as $m$ cannot be negative.
Hence, option B is correct.

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