SOLUTION: in a class of 300 students, 50 students offered maths, 100 students offered chemistry, and 200 offered physics, of 30 students offered physics and chemistry, 20 students offered ph

Question 1041204: in a class of 300 students, 50 students offered maths, 100 students offered chemistry, and 200 offered physics, of 30 students offered physics and chemistry, 20 students offered physics and maths, and 15 student offered maths and chemistry. draw a ven diagram to represent the information, how many students offered three courses?
Answer by ikleyn(52794) (Show Source): You can put this solution on YOUR website!
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in a class of 300 students, 50 students offered maths, 100 students offered chemistry, and 200 offered physics,
30 students offered physics and chemistry, 20 students offered physics and maths, and 15 student offered maths and chemistry.
draw a diagram to represent the information, how many students offered three courses?
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Let A be a finite set,
let M, C and P are  three subsets of the set A that cover A : A = M U C U P,
as it is in  our case ( subset M = {students studied Math}, C = {students studied Chem}, P = {students studied Phys} ).

Let MC is the intersection M and C (= students studied M and C)
    MP is the intersection M and P (= students studied M and P),
    CP is the intersection C and P (= students studied C and P).

Let MCP is the intersection of M, C and P (studied all 3 subjects).

Let us denote as |X| the number of elements of any subset X of A.

Then (! memorize this remarkable formula !!)

|A| = |M| + |P| + |C| - |MC| - |MP| - |CP| + |MCP|.

For the proof of this formula see the lesson Advanced problems on counting elements in sub-sets of a given finite set in this site.

Now apply the formula. You are given all numbers except of |MCP|, which is under the question.

Now you can easily find it:

|MCP| = 300 - 50 - 100 - 200 + 30 + 20 + 15 = 15.

Answer.  15 students study all 3 subjects.

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