Q70 / 150 · MHT CET 2023 (13 May Shift 2) · Mathematics

Five students are selected from n students such that the ratio of number of ways in which 2 particular students are selected to the number of ways 2 particular students not selected is $2: 3$. Then the value of $\mathrm{n}$ is

  • A
    5
  • B
    6
  • Correct answer
    C
    11
  • D
    not possible

Solution

Five students are selected from $\mathrm{n}$ students. Number of ways in which 2 particular students are sélected $=\mathrm{n}^{-2} \mathrm{C}_3$ Number of ways in which 2 particular students are not selected $={ }^{\mathrm{n}-2} \mathrm{C}_5$ $\therefore \quad$ According to the given condition, $\frac{{ }^{n-2} C_3}{{ }^{n-2} C_5}=\frac{2}{3}$ $\begin{aligned} & \Rightarrow \frac{(\mathrm{n}-2) !}{3 !(\mathrm{n}-5) !} \times \frac{5 !(\mathrm{n}-7) !}{(\mathrm{n}-2) !}=\frac{2}{3} \\ & \Rightarrow(\mathrm{n}-5)(\mathrm{n}-6)=30 \\ & \Rightarrow \mathrm{n}=11\end{aligned}$

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

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