Question

In a survey of 400 students of a college it was found that 240 study mathematics, 180 study Statistics and 140 study Accounts, 80 study mathematics and Statistics, 60 study Statistics and Accounts, 100 study Accounts and mathematics and 40 study none of these subjects. On the basis of this information answer the following questions.

Find the number of students who study at least one of these subjects.
Find the number of students who study all of these subjects

Answer 1

(a) The number of students who study at least one of these subjects is 360.

(b) The number of students who study all of these subjects is 40.

Step-by-step explanation:

Let M, S and A represents the following events.

M = Mathematics

S = Statistics

A = Accounts

Number of student = 400

240 study mathematics : n(M) = 240

180 study Statistics : n(S) = 180

140 study Accounts: n(A) = 140

80 study mathematics and Statistics : n(M∩S) = 80

60 study Statistics and Accounts : n(S∩A) = 60

100 study Accounts and mathematics : n(A∩M) = 100

40 study none of these subjects : n(M'∩S'∩A') = 40

(a)

We need to find the number of students who study at least one of these subjects.

Students who study at least one of these subjects = Total students - Number of students who study none of these subjects.

$n(M\cup S\cup A)=400-n(M'\cap S'\cap A')$

$n(M\cup S\cup A)=400-40$

$n(M\cup S\cup A)=360$

Therefore, the number of students who study at least one of these subjects is 360.

(b)

We need to find the number of students who study all of these subjects.

$n(M\cup S\cup A)=P(M)+P(S)+P(A)-n(M\cap S)-n(S\cap A)-n(M\cap A)+n(M\cap S\cap A)$

Substitute the given values.

$360=240+180+140-80-60-100+n(M\cap S\cap A)$

$360=320+n(M\cap S\cap A)$

Subtract both sides by 320.

$360-320=n(M\cap S\cap A)$

$40=n(M\cap S\cap A)$

Therefore, the number of students who study all of these subjects is 40.

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