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Q.
Number of permutations of $10$ different objects taken all at a time in which particular $4$ never come together, is
Permutations and Combinations
Solution:
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
$= 10! - 7! \times 4!$