Question

3) In a group of 50 students, 20 study subject A, 25 study subject B and 20 study subject C. 10 study both A and B, 5 study both B and C, 7 study both A and C. 2 study all three subjects. (a) show this information on Venn diagram (b) find the number of students who study (1) A only (ii) B or C (iii) A but not C (iv) none of A, B or C.​

Answer 1

Answer:

the answer is

1. 3 students
2. 47 students
3. 13 students
4. 0 students

Answer 2

the number of students who study:

1. A only  - 5 students

2. B or C  - 40 students

3. A but not C - 13 students

4. none of A, B or C - 5 students

Given:

total number of students =50

⇒ n(U)=50

number of students who study subject A = 20

⇒n(A)=20

number of students who study subject B = 25

⇒n(B)=25

number of students who study subject C = 20

⇒n(C)=20

number of students who study both subjects A and B = 10

⇒n(A∩B)=10

number of students who study subjects B and C = 5

⇒n(B∩C)=5

number of students who study subjects A and C = 7

⇒n(A∩C)=7

number of students who study all three subjects = 2

⇒n(A∩B∩C)=2

to find:

(1) A only

(ii) B or C

(iii) A but not C

(iv) none of A, B or C

solution:

n(A)=20

∴ a+b+d+e = 20  ----------(1)

n(B)=25

∴ b+c+e+f = 25    -----------(2)

n(C)=20

∴ d+e+f+g = 20   ------------(3)

n(A∩B) =10

∴ b+e = 10      ----------(4)

n(B∩C) = 5

∴ e+f = 5     -------------(5)

n(A∩C) = 7

∴ d+e = 7    -------------(6)

n(A∩B∩C) = 2

∴ e = 2     -------------(7)

from equation (4)

b+e = 10

⇒b+2 =10

b = 8

from equation (5)

e+f = 5

⇒2+f = 5

f = 3

from equation (6)

d+e= 7

⇒d+2 = 7

d=5

now from equation (1)

a+b+d+e =20

⇒a+8+5+2=20

⇒a = 20-15

⇒a = 5

∴ student who study only A ( i.e. a) = 5 students

ii) B or C

using the equation,

n(B∪C) =  n(B)+n(C) - n(B∩C)

putting the values in the equation n(B)=25, n(C)=20, and n(B∩C)=5

⇒n(B∪C) = 25 + 20 - 5

⇒n(B∪C) = 40

∴ the number of students who study B or C is 40 students.

(iii) A but not C (i.e. n(A - C))

using the equation,

n(A-C) = n(A) - n(A∩C)

putting the values in equation n(A)=20, n(A∩C)=7

⇒n(A -C)=20 - 7

⇒n(A - C) = 13

∴the students who study A but not C 13 students.

(iv) none of A, B or C

using the equation,

n(A∪B∪C) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)

putting the values in the equation,

⇒n(A∪B∪C) = 20+25+20-10-5-7+2

⇒n(AUBUC) = 65 - 20

⇒n(AUBUC)=45

n(AUBUC)' = n(U)-n(AUBUC)

⇒n(AUBUC)' = 50 - 45

⇒n(AUBUC)' =5

∴ the number of students who study none of A, B or C 5 students.

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