adjustment of Prof.,s in Round Table In How many ways can $5$ Professors of Physics Including Prof. Hardy and $3$ Three Professors of Chemistry Including Prof. Julian be seated on a Round table, If Prof. hardy and Prof. Julian are not adjacent. $\underline{\bf{My\;\;Try:}}$ First we will adjust $4$ Prof. of Physics and $2$ Prof. of Chemistry, which can be done in $6!$ ways. Now we will adjust Hardy and Julian so that they are not adjacent. I Did not Not Understand How can I adjust These two Professors. Help Required. Thanks

Since rotating the people, by convention, yields the "same" arrangement, we can assume that Hardy, who just won the Nobel Prize, sits on the one throne among the chairs.

Then Julian selects one of the chairs away from Hardy. She has $5$ choices. For each of these $5$ choices, the remaining $6$ people can be arranged in the remaining $6$ chairs in $6!$ ways, for a total of $(5)(6!)$.