In how many ways can 12 gentlemen sit around | vedclass

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Question

(D) To arrange $12$ gentlemen around a round table such that $3$ specified gentlemen are always together,we treat the $3$ specified gentlemen as a sin

Vedclass · Accepted Answer

$3! 	imes 9!$

Vedclass · Answer

(D) To arrange $12$ gentlemen around a round table such that $3$ specified gentlemen are always together,we treat the $3$ specified gentlemen as a single unit.Now,we have $(12 - 3 + 1) = 10$ units to arrange in a circle.The number of ways to arrange $n$ items in a circle is $(n - 1)!$.Thus,$10$ units can be arranged in $(10 - 1)! = 9!$ ways.Within the single unit,the $3$ specified gentlemen can be arranged among themselves in $3!$ ways.Therefore,the total number of ways is $3! 	imes 9!$.

In how many ways can $12$ gentlemen sit around a round table so that three specified gentlemen are always together?

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