Question 1041204
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in a class of 300 students, 50 students offered maths, 100 students offered chemistry, and 200 offered physics, 
{{{highlight(cross(of))}}} 30 students offered physics and chemistry, 20 students offered physics and maths, and 15 student offered maths and chemistry. 
draw a {{{highlight(Venn)}}} diagram to represent the information, how many students offered three courses?
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<pre>
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 <A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A> 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.

<U>Answer</U>.  15 students study all 3 subjects.
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