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