In how many ways can 5 professors of physics and 3 professors of chemistry be seated on a round table if
(i) Professor A.K. Goel and Dr. Agrawal are not adjacent ?
(ii) No professors of chemistry are adjacent ?

Explanation

Let `P_(1),P_(2),P_(3),P_(4) ,P_(5) ` be 5 professors of physics and `C_(1),C_(2),C_(3)` be professors of chemistry , without any restriction they can be seated on a round table in 7! Ways.
(i) Let professor Goel `(P_(1))` and professor Agrawal `(C_(1))` do not sit, adjacent to one another . Then from above illustration, the total number of seating arrangement =(7-2)6!=5(6)!=3600 ways

(ii) Let 5 professors of physics occupy their seats on the table they can be seated in `(5-1)!=4!""=24` ways. When they have been seated in any one way there remain 5 positions (x) between them. `C_(1)` can occupy any of these vacant positions in 5 ways. When `c_(1)` has occupied his position 4 places are left so `C_(2)` can occupy his position in 4 ways corresponding to each position of `C_(1)` . Then 3 position remain vacant for `C_(3)`.Hence from multiplication principle the total number of ways required =`4!xx5xx4xx3=1440`