9.Find the number of ways of seating 5 Indians, 4 Americans and 3 Russians at a round table so that 

i) all Indians sit together
ii) no two Russians sit together
iii) Persons of same nationality sit together

No. of Indians = 5
No. of Americans = 4
No. of Russians = 3
(i) All Indians are sit together:-
Treat the 5 Indians as one unit then we have 4 Americans +3 Russians +1 unit of Indians =8 units They can be arranged at a round table in (8-1)!=7! ways Now the 5 Indians among them selves can be arranged in 5 ! ways
No. of arrangements = 7 ! × 5!

(ii) No. two Russians sit together:-
First we arrange the 5 Indians + 4 Americans around the table in (9 –1)!= 8! ways Now there are 9 gaps in among these 9 persons The 3 Russians can be filled in these 9 gaps in ⁹P₃ ways 
$\therefore$ Req. no. of arrangements =$=8!\times {}^9{P}_3$

(iii) Persons of same nationality sit together:-
Treat the 5 Indians as one unit, the 4 Americans as one unit and the 3 Russians as one unit These 3 units can be arranged at round table in (3 – 1)!=2! ways Now the 5 Indians among themselves can be arranged in 5!ways Similarly the 4 Americans in 4! and 3 Russians in 3 ! ways
$\therefore$ Req. no. of arrangements
= 2! × 5! × 4! × 3!