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,
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.

Solution:

(c) 6

According to the question:

${ }^{n} P_{4}=12 \times{ }^{n} P_{2}$

$\Rightarrow \frac{n !}{(n-4) !}=12 \times \frac{n !}{(n-2) !}$

$\Rightarrow \frac{(n-2) !}{(n-4) !}=12$

$\Rightarrow(n-2)(n-3)=4 \times 3$

$\Rightarrow n-2=4$

$\Rightarrow n=6$