Tamil Nadu Board of Secondary EducationHSC Science Class 11

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 - Mathematics

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Sum

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 = 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 × 5C

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 × 5C

= `(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

Concept: Combinations
  Is there an error in this question or solution?
Chapter 4: Combinatorics and Mathematical Induction - Exercise 4.3 [Page 187]

APPEARS IN

Tamil Nadu Board Samacheer Kalvi Class 11th Mathematics Volume 1 and 2 Answers Guide
Chapter 4 Combinatorics and Mathematical Induction
Exercise 4.3 | Q 17 | Page 187

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