70 students are stydying physics mathematics and chemistry. 40 students study mathematics, 35 study physics and 30 students students chemistry.15 students are studying all the subjects. How many students are studying exactly two of the subjects?

40+35+30 - x + 15 = 70

70

To find the number of students studying exactly two subjects, we need to use the principle of inclusion-exclusion.

The principle of inclusion-exclusion is a counting technique used to find the size of the union of multiple sets. It can be expressed using the formula:

|A U B U C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

In this case, let's represent the subjects as sets:

A = Set of students studying mathematics
B = Set of students studying physics
C = Set of students studying chemistry

We are given the following information:

|A| = 40 (students studying mathematics)
|B| = 35 (students studying physics)
|C| = 30 (students studying chemistry)
|A ∩ B ∩ C| = 15 (students studying all three subjects)

Now we can substitute the values into the formula and solve:

|A U B U C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

|A U B U C| = 40 + 35 + 30 - |A ∩ B| - |B ∩ C| - |C ∩ A| + 15

Since we want to find the number of students studying exactly two subjects, we need to subtract the number of students studying all three subjects from the above expression:

Number of students studying exactly two subjects = |A U B U C| - |A ∩ B ∩ C|

Number of students studying exactly two subjects = (40 + 35 + 30 - |A ∩ B| - |B ∩ C| - |C ∩ A| + 15) - 15

Now we need to determine the values of |A ∩ B|, |B ∩ C|, and |C ∩ A|. Unfortunately, these values are not given directly. However, we can find them using the given information.

Using the principle of inclusion-exclusion, we know that:

|A ∩ B ∩ C| = |A| + |B| + |C| - |A U B| - |A U C| - |B U C| + |A U B U C|

Substituting the given values:

|A ∩ B ∩ C| = 40 + 35 + 30 - |A U B| - |A U C| - |B U C| + 15

Since |A U B U C| = 15, we can rewrite the equation as:

15 = 40 + 35 + 30 - |A U B| - |A U C| - |B U C| + 15

Simplifying the equation:

0 = 40 + 35 + 30 - |A U B| - |A U C| - |B U C|

Now we know that |A ∩ B ∩ C| = 15 and |A U B U C| = 15, which means |A U B| + |A U C| + |B U C| must be equal to 70 - 15 = 55.

Let's assume the number of students studying exactly two subjects is x.

Using the principle of inclusion-exclusion, we can express |A U B U C| as:

|A U B U C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|

Substituting the given values:

15 = 40 + 35 + 30 - |A ∩ B| - |B ∩ C| - |C ∩ A| + 15

15 = 40 + 35 + 30 - |A ∩ B| - |B ∩ C| - |C ∩ A| + 15

15 = 100 - |A ∩ B| - |B ∩ C| - |C ∩ A|

|A ∩ B| + |B ∩ C| + |C ∩ A| = 100 - 15

|A ∩ B| + |B ∩ C| + |C ∩ A| = 85

Since |A ∩ B ∩ C| = 15, we can rewrite the equation as:

15 + x = 85

x = 85 - 15

x = 70

Therefore, the number of students studying exactly two subjects is 70.

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