Question 1201973
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In survey of 900 workers 600 owned houses, 500 owned cars and 345 owned {{{highlight(cross(boat))}}} <U>boats</U>, 
300 owned cars and houses,250 owned houses and boats and 270 owned cars and {{{highlight(cross(boat))}}} <U>boats</U>.
200 owned the three. 
(a) Find how many workers don't owned any of items.
(b) Find how many workers owned only two of the items.
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<pre>
(a)  You are given a universal set of 900 workers and three its basic subsets

         - H (owned houses) of 600 workers;
         - C (owned cars)   of 500 workers;
         - B (owned boats)  of 345 workers.

     You also are given the info about in-pair intersections of these three subsets

         - CH (owned cars   and houses)  of 300 workers;
         - HB (owned houses and boats)   of 250 workers;
         - CB (owned cars   and boats)   of 270 workers.

     You also are given the info about the triple intersections of these three subsets

         - CHB (owned all three items)  of 200 workers.


     Use the Inclusion-Exlusion principle.  The Inclusion-Exclusion formula is

         n(H U C U B) = n(H) + n(C) + n(B) - n(HC) - n(HB) - n(CB) + n (CHB) = 

                      = 600 + 500 + 345 - 300 - 250 - 270 + 200 = 825.

     Thus 825 workers own at least one of three items.

     Hence, the rest  1000-825 = 175  workers don't have any of items.    <U>ANSWER</U>



(b)  To find the number of workers who own only two of the items, subtract triple intersection from 
     the corresponding in-pair intersection.

     You will get:

         n(CH_only) = n(CH) - n(CHB) = 300 - 200 = 100  (owned cars   and houses only);

         n(HB_only) = n(HB) - n(CHB) = 250 - 200 =  50  (owned houses and boats only);

         n(CB_only) = n(CB) - n(CHB) = 270 - 200 =  70  (owned cars   and boats only).

     These sets CH_only, HB_only and CB_only are disjoint (i.e. have empty intersection;  THEREFORE

     the number of workers owned only two of the items is the sum  100 + 50 + 70 = 220.    <U>ANSWER</U>
</pre>

Solved.


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To see many other similar &nbsp;(and different) &nbsp;solved problems on Inclusion-Exclusion principle, &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.



On Inclusion-Exclusion principle, &nbsp;see this Wikipedia article


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