If in a group of n distinct objects, the number of arrangements of 4 o

1.	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» Option : (C) In a group of n distinct objects, The number of arrangements of 4 objects is = ⁿP₄ and the number of arrangements of 2 objects is = ⁿP₂. According to the given problem, \(P^n_4\) = 12 x \(P^n_2\) ⇒ \(\frac{n!}{(n-4)!}\) = 12 x \(\frac{n!}{(n-2)!}\) ⇒ n×(n-1)×(n-2)×(n-3) = 12×n×(n-1) ⇒ (n-2)×(n-3) =12 = (6-2)(6-3) ⇒ n = 6 Hence, The number of objects = 6. [Note : \(P^n_4\) = ⁿP₄ and \(P^n_2\) = ⁿP₂]

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»

Option : (C)

In a group of n distinct objects,

The number of arrangements of 4 objects is = ⁿP₄ and the number of arrangements of 2 objects is = ⁿP₂.

According to the given problem,

\(P^n_4\) = 12 x \(P^n_2\)

⇒ \(\frac{n!}{(n-4)!}\) = 12 x \(\frac{n!}{(n-2)!}\)

⇒ n×(n-1)×(n-2)×(n-3) = 12×n×(n-1)

⇒ (n-2)×(n-3) =12 = (6-2)(6-3)

⇒ n = 6

Hence,

The number of objects = 6.

[Note : \(P^n_4\) = ⁿP₄ and \(P^n_2\) = ⁿP₂]