Q166 / 270 · NDA 2019 (Phase 1) · Mathematics

Paragraph:In a school, all the students play at least one of three indoor games - chess, carrom and table tennis, 60 play chess, 50 play table tennis, 48 play carrom, 12 play chess and carrom, 15 play carrom and table tennis, 20 play table tennis and chess. Question:What can be the minimum number of students in the school?

  • A
    123
  • Correct answer
    B
    111
  • C
    95
  • D
    63

Solution

Number of students who play chess, $\mathrm{n}(\mathrm{A})=60$ Number of students who play tennis, $n(B)=50$ Number of students who play carrom, $n(C)=48$ Given, $\mathrm{n}(\mathrm{A} \cap \mathrm{B})=20$ $\mathrm{n}(\mathrm{B} \cap \mathrm{C})=15$ $\mathrm{n}(\mathrm{A} \cap \mathrm{C})=12$ $\begin{aligned} & \mathrm{n}(\mathrm{A} \cup \mathrm{B} \cup \mathrm{C})=\mathrm{n}(\mathrm{A})+\mathrm{n}(\mathrm{B})+\mathrm{n}(\mathrm{C})-\mathrm{n}(\mathrm{A} \cap \mathrm{B})- \\ & \mathrm{n}(\mathrm{B} \cap \mathrm{C})-\mathrm{n}(\mathrm{A} \cap \mathrm{C})+\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}) \\ &= 60+50+48-20-15-12+\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}) \\ &= 111+\mathrm{n}(\mathrm{A} \cap \mathrm{B} \cap \mathrm{C})\end{aligned}$ So, minimum number of students $=111$

Hence, the correct answer is option: 'B'.

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