Q56 / 160 · AP EAMCET 2023 (18 May Shift 1) · Mathematics

The least value of $n$ for which ${ }^{(n-1)} C_2+{ }^{(n-1)} C_3>{ }^n C_2$ is

  • A
    $7$
  • B
    $4$
  • C
    $5$
  • Correct answer
    D
    $6$

Solution

${ }^{n-1} C_2+{ }^{n-1} C_3>{ }^n C_2$ ${ }^n C_3>{ }^n C_2 \Rightarrow \frac{n !}{3 !(n-3) !}>\frac{n !}{2 !(n-2) !}$ $\Rightarrow \frac{1}{3} \geq \frac{1}{(n-2)}$ $\begin{aligned} & \Rightarrow \mathrm{n}-2>3 \Rightarrow \mathrm{n}>5 \\ & \Rightarrow \mathrm{n}=6\end{aligned}$

Hence, the correct answer is option: 'D'.

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