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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

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#### Solution

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 = 7C_{3}

The number of ways of selecting 2

Americans from 5 Americans is = 5C_{2}

Total number of ways in this case is = 7C_{3} × 5C_{2}

**Case (ii):** 4 Indians and 1 American

The number of ways of selecting 4

Indians from 7 Indians is = 7C_{4}

The number of ways of selecting 1

American from 5 Americans is = 5C_{1}

The total number of ways, in this case, is = 7C_{4} × 5C_{1 }

**Case (iii):** 5 Indians no American

Number of ways of selecting 5

Indians from 7 Indians is = 7C_{5}

Total number of ways, in this case, = 7C_{5} × 5C_{0}

∴ Total number of ways of forming the committee

= 7C_{3} × 5C_{2} + 7C_{4} × 5C_{1} + 7C_{5} × 5C_{0 }

= `(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

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