Question

If in a group of n distinct objects, the number of arrangements of 4 objects is 12 times the number of arrangements of 2 objects, then the number of objects is
(a) 10
(b) 8
(c) 6
(d) none of these.

Answer 1

(c) 6
According to the question:
ⁿP₄ = 12$\times$ⁿP₂
$$\begin{gathered} \Rightarrow \frac{n!}{\left(n-4\right)!}=12\times \frac{n!}{\left(n-2\right)!} \\ \Rightarrow \frac{\left(n-2\right)!}{\left(n-4\right)!}=12 \\ \Rightarrow \left(n-2\right)\left(n-3\right)=4\times 3 \\ \Rightarrow n-2=4 \\ \Rightarrow n=6 \end{gathered}$$