SOLUTION: If n(A)=15, n(B)=18, n(C)=16, n(A∩B)=7, n(A∩C)=6, n(B∩C)=7, and n(A∪B∪C)=30 find n(A∩B∩C). n(A∩B∩C)=9 n(A∩B∩

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Question 1055594: If n(A)=15, n(B)=18, n(C)=16, n(A∩B)=7, n(A∩C)=6, n(B∩C)=7, and n(A∪B∪C)=30 find n(A∩B∩C).
n(A∩B∩C)=9
n(A∩B∩C)=1
n(A∩B∩C)=2
n(A∩B∩C)=17
n(A∩B∩C)=3
Please explain. Thank you
Answer by ikleyn(52777)   (Show Source): You can put this solution on YOUR website!
.
From the elementary set theory, there is the formula


n(A∪B∪C) = n(A) + n(B) + n(C) - n(A∩B) - n(A∩C) - n(B∩C) + n(A∩B∩C).   (1)


or, equivalently, 

n(A∩B∩C) = n(A∪B∪C) - n(A) - n(B) - n(C) + n(A∩B) + n(A∩C) + n(B∩C).   (2)


Therefore,  n(A∩B∩C) = 30 - 15 - 18 - 16 + 7 + 6 + 7 = 1. 


Regarding the explanations, read the lesson

    - Advanced problems on counting elements in sub-sets of a given finite set

in this site.


Everything is explained there.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic "Miscellaneous word problems".


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