Question

Find the number of ways of forming a committee of 5 members out of 7 Indians and 5 Americans, so that always Indians will be the majority in the committee

Answer 1

Number of Indians = 7

Number of Americans = 5

Number of members in the committee = 5

Selection of 5 members committee with majority Indians

Case (i): 3 Indians and 2 Americans

The number of ways of selecting 3

Indians from 7 Indians is = 7C₃

The number of ways of selecting 2

Americans from 5 Americans is = 5C₂

Total number of ways in this case is = 7C₃ × 5C₂

Case (ii): 4 Indians and 1 American

The number of ways of selecting 4

Indians from 7 Indians is = 7C₄

The number of ways of selecting 1

American from 5 Americans is = 5C₁

The total number of ways, in this case, is = 7C₄ × 5C₁ 

Case (iii): 5 Indians no American

Number of ways of selecting 5

Indians from 7 Indians is = 7C₅

Total number of ways, in this case, = 7C₅ × 5C₀

∴ Total number of ways of forming the committee

= 7C₃ × 5C₂ + 7C₄ × 5C₁ + 7C₅ × 5C₀ 

= `(7!)/(3!(7 - 3)!) xx (5!)/(2!(5 - 2)!) + (7!)/(4!(7 - 2)!) xx 5 + (7!)/(4!(7 - 2)!) xx 1`

= `(7!)/(3!  4!) xx (5!)/(2!  3!) + (7!)/(4!  3!) xx 5 + (7!)/(5!  2!)`

= `(7 xx 6 xx 5 xx 4!)/(3! xx 4!) xx (5 xx 4 xx 3!)/(2! xx 3!) + (7 xx 6 xx 5 xx 4!)/(4! xx 3!) xx 5 + (7 xx 6 xx 5!)/(5! xx 2!)`

= `(7 xx 6 xx 5)/(3 xx 2  xx 1) xx (5 xx 4)/(2 xx 1) + (7 xx 6 xx 5 xx 5)/(3 xx 2 xx 1) + (7 xx 6)/(2 xx 1)`

= 7 × 5 × 5 × 2 + 7 × 5 × 5 + 7 × 3

= 350 + 175 + 21

= 546