the following analysis shows how students of various department for 2018/2019 academic session, 10 for economics only, 150 from economics and geography. 90 from economics but not political science, 210 from geography, 120 fron political science and geography, 180 from economics. if the total of student registered for courses is 250, determine the no of students who registered for (i)political science (ii)political science and geography but not economics (iii)economics and political science but not geography

(i) The number of students who registered for political science is 150

(ii) The number of students who registered for political science and geography but not economics is 50

(iii) The number of students who registered for economics and political science but not geography is 20

The Venn diagram for the question is shown in the attachment below.

E represents Economics

G represents Geography

and P represents Political science

From the question,

10 for economics only, that is, n(E∩G'∩P') = 10

150 from economics and geography, that is n(E∩G) = 150

90 from economics but not political science, that is, n(E∩P') = 90

210 from geography, that is, n(G) = 210

120 from political science and geography, n(P∩G) = 120

180 from economics, that is, n(E) = 180

Total of student registered for courses is 250, that is, n(ξ) = 250

Now from the given information,

n(E∩G∩P') = n(E∩P') - n(E∩G'∩P') = 90 - 10 = 80

n(E∩G∩P) =  n(E∩G) - n(E∩G∩P') = 150 - 80 = 70

n(E∩P∩G') = n(E) - [n(E∩G'∩P') + n(E∩G∩P') + n(E∩G∩P)]

= 180 - [10+80+70] = 180 - 160 = 20

n(P∩G∩E') = n(P∩G) - n(E∩G∩P) = 120 - 70 = 50

n(G∩E'∩P') = n(G) - [n(E∩G∩P') + n(E∩G∩P) + n(P∩G∩E')]

= 210 -[80+70+50] = 210 - 200 = 10

All of these are shown in the Venn diagram

(i) To determine the number of students who registered for political science, we will first determine n(P∩G'∩E'), that is, those who registered for political science only.

Let n(P∩G'∩E') = x

Then, using the Venn diagram, we can write that

10 + 80 + 20 + 70 + 10 + 50 + x = n(ξ) = 250

240 + x = 250

x = 250 - 240

x = 10

n(P∩G'∩E') = 10

∴ 10 registered for political science only

Now, number of students who registered for political science n(P) is

n(P) = n(E∩P∩G') + n(E∩G∩P) +  n(P∩G∩E')  + n(P∩G'∩E')

n(P) = 20 + 70 + 50 + 10

n(P) = 150

∴ 150 students registered for political science

(ii) For the number of students who registered for political science and geography but not economics, that is, n(P∩G∩E')

As determined above and as shown in the Venn diagram,

n(P∩G∩E') = 50

∴ 50 students registered for political science and geography but not economics

(iii) For the number of students who registered for economics and political science but not geography, that is, n(E∩P∩G')

As determine above and as shown in the Venn diagram as well

n(E∩P∩G') = 20

∴ 20 students registered for economics and political science but not geography

Hence,

(i) The number of students who registered for political science is 150

(ii) The number of students who registered for political science and geography but not economics is 50

(iii) The number of students who registered for economics and political science but not geography is 20

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