Question

Find the number of ways of forming a Committee of $5$ members, out of $6$ Indians and $5$ Americans, so that always the Indians will be in majority in the Committee.

Answer 1

Total $5$ members possible ways that indians are in majority when there are $6$ Indians and $5$ americans

Case 1: $5Indians+0Americans$

Case 2: $4Indians+1Americans$

Case 3: $3Indians+2Americans$

So for Case 1 ways $={}^6{C}_5{\times }^5{C}_0$

So for Case 2 ways $={}^6{C}_4{\times }^5{C}_1$

So for Case 3 ways $={}^6{C}_3{\times }^5{C}_2$

So totally $={}^6{C}_5{\times }^5{C}_0{+}^6{C}_4{\times }^5{C}_1{+}^6{C}_3{\times }^5{C}_2$

$=6+15\times 5+20\times 10=281$ways