Question 1195171
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<pre>

You are given the universal set U of 43 cars and 3 its basic subsets R, A and H 
(see the table below).

    U    43    total cars

    R    23    Red cars

    A    25    cars with Air conditioning

    H    29    cars with Heated seats


Also, you are given info about their in-pair intersections and about their triple intersection.


    RA   17    Red cars with Air conditioning

    RH   15    Red cars with Heated seats

    AH   16    cars with Air conditioning and Heated seats

    RAH  10    Red cars with Air conditioning and Heated seats



Having this info well organized, you can easily answer all questions (b), (c) and (d) 
without drawing Venn diagram.



(b)  The set {Red cars had neither Air Conditioning nor Heated Seats} is  R \ ((RA U RH) \ RAH)   

     and their number is  23 - (17 + 15 - 10)  = 1.    <U>ANSWER</U>                                     



(c)  The set {cars had Air Conditioning were not red} is  A \ RA,

     and their number is  25 - 17 = 8.     <U>ANSWER</U>



(d)  The set {cars were not red and had neither Air Conditioning nor Heated Seats} is  U \ (R U A U H).
     
     So, calculate the number of cars in the union (R U A U H}  first.

     For it, use the inclusion-exclusion princuple/(formula)

         n(R U A U H) = n(R) + n(A) + n(H) - n(RA) - n(RH) - n(AH) + n(RAH) = 

                      =  23  +  25  +  29  -  17   -  15   -  16   +  10    = 39.


     Now the last step gives the answer to question (d) :  

        the number of cars were not red and had neither Air Conditioning nor Heated Seats = 43 - 39 = 4.    <U>ANSWER</U>
</pre>

Solved.


----------------


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 (!)