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(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
Learn more here: brainly.com/question/15311191
