UPSC NDA I 2019 Mathematics Solved Paper

Concept:Use Venn diagram and inclusion-exclusion to find the minimum total number of students by minimizing the overlap region common to all three games.Explanation:Let $a$, $b$, $c$, $d$, $e$, $f$, $g$ represent the number of students in the seven disjoint regions of the Venn diagram for chess, carom, and table tennis.Given: $60$ play chess, $50$ play table tennis, $48$ play carom.Also: $12$ play chess and carom, $15$ play carom and table tennis, $20$ play table tennis and chess.Write equations:$a + b + d + e = 60$ (chess)$d + e + f + g = 50$ (table tennis)$b + c + e + f = 48$ (carom)$b + e = 12$ (chess and carom)$e + f = 15$ (carom and table tennis)$d + e = 20$ (table tennis and chess)Add the first three equations: $(a+b+d+e)+(d+e+f+g)+(b+c+e+f)=60+50+48=158$.Subtract the last three equations: $-(b+e)-(e+f)-(d+e)=-(12+15+20)=-47$.Result: $a + b + c + d + f + g = 158 - 47 = 111$.Total students $= (a+b+c+d+f+g) + e = 111 + e$.To minimize the total, set $e = 0$ (no student plays all three games).Thus, minimum number of students $= 111$.Answer:$111$