Exercise2.3

(7) Find the number of ways of sitting 5 Indians , 4 Americans, 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 sit together
Consider 5 Indians as one unit
(5 indians ) 4 Americans , 3 Russians (1+4+3=8 
can be arranged around a table in $\left(8-1\right)!$
5 indians can be arranged in 5! ways 
no.of permutations =5! x 7!
(ii) no two Russians sit together
arrange 5 indians , 4 Americans around a table in (9-1)!=8!
there 9 place between these 9 persons and 3 Russians can be arranged in ${}^{9}P_3$ ways
(iii) persons of same nationality sit together
denote  5 Indians as I
              4  Americans A
              3 Russians R
I,A,R can be arrangeed around a circle in (1-3)! = 2! ways
5 Indians can be rearranged in 5! ways
4 Americans can be rearranged in 4! ways
3 Russians can be rearranged in 3! ways
no.of permutations $2!\times 5!\times 4!\times 3!$