Question

Number of permutations of $10$ different objects taken all at a time in which particular $4$ never come together, is

• A. $10!\times 4!$
• B. $10!-4!$
• C. $\frac{7!6!}{4!}$
• D. $10!-7!4!$

Answer 1

Answer: (D)

Fact. Number of permutations of $n$ things taken all at a time = Number of ways of particular things arc always together + particular things are never together.
Let $4$ particular things denoted by $k_1,k_2,k_3,k_4$ arc as one thing.
$\therefore$ Total permutations = particular things always together + never together
$\Rightarrow 10!=(10-4+1)!4!+$ Never together
$=7!4!+$ Never together
$\therefore$ Particular $4$ things never come together
$=10!-7!\times 4!$