Question 1190277
.
A survey of a group of 112 tourists was taken in St Louis, the survey showed the following:
62 the {{{highlight(cross(Taurus))}}} <U>tourists</U> plan to visit gateway Arch;
50 plan to visit the zoo;
11 plan to visit the art museum and the zoo, but not the gateway Arch;
12 plan to visit the art museum and the gateway arch, but not the zoo;
19 plan to visit the gateway Arch and the zoo, but not the art museum;
8 plan to visit the art museum, the zoo, and the gateway Arch;
14 plan to visit none of the three places.
How many plan to visit the art museum only?
~~~~~~~~~~~~~~


<pre>
We have a universal set U of 112 tourists (total) and 3 its subsets

    A wishing to visit Arch       ( n(A) = 62 )

    M wishing to visit Museum     ( n(M) = ? )

    Z wishing to visit Zoo.       ( n(Z) = 50 )


We know n(A) = 62,

        n(Z) = 50.


We have some info about in-pair intersections of these subsets AM, AZ and MZ, as well as about 
their triple intersection AMZ.  This info is

        n(MZ \ AMZ) = 11,

        n(AM \ AMZ) = 12,

        n(AZ \ AMZ) = 19,

        n(AMZ) = 8.


From this info we have  n(MZ) = 11+8 = 19;  n(AM) = 12+8 = 20;  n(AZ) = 19+8 = 27.


Having it, we can apply the inclusive-exclusive formula (principle), which says

    n(A U M U Z) = n(A) + n(M) + n(Z) - n(AM) - n(AZ) - n(MZ) + n(AMZ).


Next, substitute what you just know into the formula. Notice that n(A U M U Z) = 112-14 = 98.


At this moment, we know all the terms in this formula, except of n(M)

    98 = 62 + n(M) + 50 - 20 - 19 - 27 + 8,   or  98 = 54 + n(M).


From the last equation, we get  n(M) = 98 - 54 = 44.


The problem asks us about {{{n(M[only])}}},  which is 

    {{{n(M[only])}}} = n(M) - ( n(AM) + n(MZ) - n(AMZ) ) = 44 - (20 + 19 - 8) = 13.


<U>ANSWER</U>.  13 tourists plan to visit the museum only.
</pre>

Solved.


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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 (!)