[Math] Largest possible number of student passed in all three subject

combinatoricsself-learning

Question:

From $50$ students taking examination in Mathematics, Physics and Chemistry, $37$ passed in mathematics, $24$ passed in physics and $43$ passed in chemistry. At most $19$ students passed in mathematics and physics, at most $29$ passed in mathematics and chemistry and at most $20$ passed in physics and chemistry.

The largest number of students passed in all the three subjects are?

My Efforts:

In this type of problem i find it easier to solve with Venn Diagram.

enter image description here

I Don't know what to do ahead.

Best Answer

Use inclusion/exclusion: $50-37-24-43+19+29+20-X=0$.

Now find $X$...

Your Venn Diagram is an equivalent solution if you put:

  • $[37,24,43]$ in the big circles
  • $[19,29,20]$ in the smaller ellipses
  • $[X]$ in the smallest intersection area
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