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15 votes

In a class of $300$ students in an M.Tech programme, each student is required to take at least one subject from the following three:

  • M600: Advanced Engineering Mathematics
  • C600: Computational Methods for Engineers
  • E600: Experimental Techniques for Engineers

The registration data for the M.Tech class shows that $100$ students have taken M600, $200$ students have taken C600, and $60$ students have taken E600. What is the maximum possible number of students in the class who have taken all the above three subjects?

  1. $20$
  2. $30$
  3. $40$
  4. $50$
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3 Answers

Best answer
17 votes
17 votes
Let the no. of students who took all courses be $x$.

Since every one must take at least 1 course, we have

$200 - x + 100 - x + 60 - x \geq 300 -x \\\implies 360-2x \geq 300 \implies x \leq 30.$

Correct Answer: $B$
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4 votes
4 votes

Let the number of students who have taken all $3$ subjects are $x$

Remaining students(Out of total $300$) = $300 - x$

Subtract $x$ from the number of students in $M600$ , $C600$, $E600$ to get the number of students who have taken either one subject or two subjects.

In $M600$ we have $100-x$ such students
In $C600$ we have $200-x$ such students
In $E600$ we have $60 - x$ such students

Now, $(100-x) + (200-x) + (60-x) \geq 300$

We get $ x \leq 30$ hence Option B is the correct answer
(p.s The above inequality will become an equality if number of students who have taken exactly $2$ subjects is $0$)

2 votes
2 votes

Total Number of student is 300.

n(M-600) = 100, n(C-600) = 200, n(E-600) = 60  (Given data in question)

We need to find max((M-600)∩(C-600)∩(E-600)).............................?

n(M-600)+n(C-600)+n(E-600) = 100+200+60 = 360 (Every one has taken at-least one subject) then 360 - 300 = 60 (Remeber)

we have 60 people who has been enrolled for more than one subject in M.tech Program.

so 60 / 2 = 30 (To maximize the 3 subject student we can enroll 30 student max for all three subject).

Please comment if it's easy to understand i will upload the pictorial representation with some small example.

Answer:

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