Question 1206204
.
A group of 2468 students were surveyed about the courses they were taking at their college with the following results:

1140 students said they were taking Psychology.
1244 students said they were taking Dance.
1110 students said they were taking History.
599 students said they were taking Psychology and Dance.
550 students said they were taking Psychology and History.
602 students said they were taking History and Dance.
349 students said they were taking all three courses.

a) How many students took Psychology, History, or Dance?


b) How many students took none of the courses?


c) How many students took Psychology or didn't take History?


d) How many students took Psychology or Dance, but not History?


e) How many students took Psychology & History or took History & Dance?


f) How many students took History and Dance, but not Psychology?
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<pre>
(a)  For (a), use the Inclusion-Exclusion principle. 

     It says that the number of elements in the union of three subset P, D and H is the sum
     of numbers of elements in the separate sets MINUS the number of element in each of their 
     in-pair intersections PLUS the number of elements in the triple intersection.

     According to this formula, n(P U H U D) = 1140 + 1244 + 1110 - 599 - 550 - 602 + 349 = 2092.

     So, the <U>ANSWER</U> to (a) is 2092.



(b)  This number is the complement of 2092 to 2468, i.e.  2468-2092 = 376.



(e)  This set is precisely the union of the three subsets P, D and H.

     The number of elements in this union we just calculated in (a): it is 2092.



(F)  It is easy.  To compute the number of these students, take the number of students in (H U D),
     it is 602 (given), and subtract from it the number of student in the triple intersection, which is 349.

     So, the difference 602 - 349 = 253 is the <U>ANSWER</U> to (f)
</pre>

I hope that from my post you will learn a lot of useful information.


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On inclusion-exclusion principle, &nbsp;see this Wikipedia article


https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle



To see many other similar &nbsp;(and different) &nbsp;solved problems, &nbsp;see the lessons


&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Advanced-probs-counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Advanced problems on counting elements in sub-sets of a given finite set</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Challenging-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson>Challenging problems on counting elements in subsets of a given finite set</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/misc/Selected-problems-on-counting-elements-in-subsets-of-a-given-finite-set.lesson>Selected problems on counting elements in subsets of a given finite set</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/Permutations/Inclusion-Exclusion-principle.lesson>Inclusion-Exclusion principle problems</A> 


in this site.



Happy learning (!)