Question

In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to _____

Answer 1

Given data:
Let:
$M =$ number of students who studied Mathematics,
$P =$ number of students who studied Physics,
$C =$ number of students who studied Chemistry.

Given conditions:
$125 \leq M \leq 130, \quad 85 \leq P \leq 95, \quad 75 \leq C \leq 90.$
Number of students studying two subjects:
$|P \cap C| = 30, \quad |C \cap M| = 50, \quad |M \cap P| = 40.$

Number of students studying none:
$|U| - |M \cup P \cup C| = 10 \implies |M \cup P \cup C| = 210.$
Using the formula for the union of three sets:
$|M \cup P \cup C| = M + P + C - |M \cap P| - |P \cap C| - |C \cap M| + |M \cap P \cap C|.$
Substituting the values: $210 = M + P + C - 40 - 30 - 50 + x,$
where $x$ is the number of students who studied all three subjects.

Simplifying: $M + P + C + x = 330.$

Finding the range for $x$:
From the given bounds: $125 \leq M \leq 130, \quad 85 \leq P \leq 95, \quad 75 \leq C \leq 90.$
Therefore: $15 \leq x \leq 30.$
Calculating $m + n$:
$m = 15, \quad n = 30.$ $m + n = 15 + 30 = 45.$