Question

In a survey of 25 students, it was found that 15
have taken Mathematics, 12 have taken Physics and
11 have taken Chemistry, 5 have taken Mathematics
and Chemistry, 9 have taken Mathematics and
Physics, 4 have taken Physics and Chemistry and 3 -
have taken all the three subjects.
Find the number who have taken
A.Physics only
B.Maths
C.Atkeast one of the three subjescts

Answer 1

☘Answer:-

Given:-

1. Total students are 25.
2. 15 have taken Mathematics.
3. 12 have taken Physics.
4. 11 have taken Chemistry.
5. 9 have taken Mathematics and physics.
6. 4 have taken Physics and chemistry.
7. 3 have taken all the three subjects.

To find:-

• A.Physics only.
• B.Maths only.
• C.Atleast one of thethreesubjescts.

Solution:-

$Let \: M \: : \: Set \: \: of \: \: students \: \: who \: \: have \: \: taken \: \: Maths \\ \\ P \: : \: Set \: \: of \: \: students \: \: who \: have \: \: taken \: \: Physics \\ \\ C \: : \: Set \: \: of \: \: students \: \: who \: \: have \: \: taken \: \: Chemistry$

$Given,$

$Total \: \: students \: \: n(U) \: = \: 25$

$n(M) \: = \: 15, n(P) \: = \: 12, n(C) \: = \: 11$

$n(M ∩ C) \: = \: 5, n(P ∩ C) \: = \: 4, n(M ∩ P) \: = \: 9$

$n(M ∩ P ∩ C) \: = \: 3$

☘$A. Physics = >$

$Number \: \: of \: \: students \: \: taking \: \: only \\ Physics \: = \: n \: (P - (M ∪ C))$

$= n(P) - n(P ∩ (M ∪ C))$

$= n(P) - [n(P ∩ M) + n(P ∩ C) - n((P ∩ M) ∩ (P ∩ C)) ]$

$= n(P) - n(P ∩ M) - n(P ∩ C) + n(P ∩ M ∩ C)$

$= 12 - 9 - 4 + 3$

$= 15 - 13$

$= 2$

☘$B. Maths = >$

$Number \: \: of \: \: students \: taking \: only \: Maths \\ \: = \: n(M - (P ∪ C))$$= n(M) - n(M ∩ (P ∪ C))$

$= n(M) - [n(M ∩ P) + n(M ∩ C) - n((M ∩ P) ∩ (M ∩ C)) ]$

$= n(M) - n(M ∩ P) - n(M ∩ C) + n(M ∩ P ∩ C)$

$= 15 - 9 - 5 + 3$

$= 18 - 14$

$= 4$

☘$C. \: Atleast \: \: one \: \: of \: \: the \: \: three \: \: subjescts = >$

$Number \: \: of \: \: students \: \: taking \: \: at \\ \: \: least \: \: one subject \: = n(M ∪ P ∪ C)$

$= n M) + n(P) + n(C) - n(M ∩ P) - n(P ∩ C) - n(M ∩ C) + n(M ∩ P ∩ C)$

$= 15 + 12 + 11 - 9 - 4 - 5 + 3$

$= 41 - 18$

= 23

✪ A. Only Physics = 2.

✪B. Only Mathematics = 4.

✪C. Atleast one of the three subjescts = 23.

Answer 2

Answer:

above answer is correct