Please help with Math

Question

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 help. Thank you

Answer 1

I solved this by making a diagram of three partially overlapping circles, labeling their circumferences A, B and C, with A at top left, B at top right, C beneath at middle (partially overlapping, remember!).

 

Inside the 7 distinctly boundaried areas, I place a lowercase letter to represent the counts for each. From left to right, along the top portion of this diagram, I used letters a, b and c.

A bit below this, at the same level as the region A∩B∩C, from left to right, I placed the letters d, x and e such that the count associated with  A∩B∩C is "x".

Then the bottom part of the diagram, in the "rest of C" which does not overlap any other circle, I placed letter f.

 

The question is equivalent to SOLVE FOR x.

I replace letter b by the expression 7-x, letter d by 6-x, and letter e by 7-x because it's given that

n(A∩B)=7, n(A∩C)=6, n(B∩C)=7.

 

Then I wrote equations for n(A), n(B) and n(C), as follows, respectively:

15 = a+7+6-x

18=c+7+7-x

16=f+7+6-x

and one more equation reflecting n(A∪B∪C):  30 = a+c+f+7+(6-x)+(7-x).

 

I have four linear equations in four unknowns; I solved them by the row-echelon method learned in Algebra II courses.

 

My answers area=3, c=5, f=4, x=1.

 

I checked these for all of the givens, and all was well.  

Final answer: x = n(A∩B∩C) = 1