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

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

BITSATBITSAT 2018

Solution:

Let us consider the $3$ specific gentlemen as a single person that occupies three times the space.
So, now we have $10$ persons who can be made to sit in $9 !$ ways and in the space for $3$ gentlemen we can permute them in $3 !$ ways.
So, total ways are $3 !\, 9 !$.