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:
${ }^{n} P_{4}=12 \times{ }^{n} P_{2}$
$\Rightarrow \frac{n !}{(n-4) !}=12 \times \frac{n !}{(n-2) !}$
$\Rightarrow \frac{(n-2) !}{(n-4) !}=12$
$\Rightarrow(n-2)(n-3)=4 \times 3$
$\Rightarrow n-2=4$
$\Rightarrow n=6$
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