Question

in a survey of 25 student it was found that 15 had taken math,12 had taken physics and 11 had taken chemistrt, 5 had taken maths and chemistry,9 had taken maths and physics,4 had taken physics and chemistry and 3 had taken all the three subject find the number of student that had taken 1) only chemistry 2) only maths 3) only physics
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Answer 1

Given :-

◉ There are total students n(U) = 25.

◉ The students are given as:

• 15 took Maths, n(M) = 15
• 12 took Physics, n(P) = 12
• 11 took Chemistry, n(C) = 11
• 5 had taken Maths and Chemistry.
• 9 had taken Maths and Physics.
• 4 had taken Physics and Chemistry.
• 3 had taken all three subjects.

To Find :-

◉ Students who took:

• Only Chemistry.
• Only Maths.
• Only Physics

Solution :-

Let the students be in a set for each subjects labelled as M(Maths) , P(Physics) and C(Chemistry)

Also, It is given that there are common students in both the sets M and C.

∴ n(M ⋂ C) = 5

Similarly,

• n(M ⋂ P) = 9
• n(P ⋂ C) = 4
• n(M ⋂ P ⋂ C) = 3

[i]

We have to find the number of students who had taken only Chemistry.

So, We would have to subtract those students from Number of Chemistry students n(C) who had taken other subjects as well.

⇒ n(C) - n(C⋂P) - n(C⋂M) + n(M ⋂ P ⋂ C)

⇒ 11 - (4 + 5 - 3)

⇒ 11 - (6)

⇒ 5

So, 5 Students had taken only Chemistry.

[ii]

We need to find students who had taken only Maths.

So, We would have to remove students which are also in other subjects as well.

⇒ n(M) - n(M⋂P) - n(M⋂C) + n(M ⋂ P ⋂ C)

⇒ 15 - (9 + 5 - 3)

⇒ 15 - (11)

⇒ 15 - 11

⇒ 4

Hence, 4 Students had taken only Maths.

[iii]

To find only the number of students who had taken only Physics, We need to subtract number of students who had taken other subjects as well.

⇒ n(P) - n(P ⋂ M) - n(P ⋂ C) + n(M ⋂ P ⋂ C)

⇒ 12 - (9 + 4 - 3)

⇒ 12 - (10)

⇒ 2

∴ 2 Students had taken only Physics.