SOLUTION: In a recent survey of 180 housewives. The following were determined with regard to their detergent soap preferences 65 use brand A 140 use brand B 145 use brand C 115 use B

                           Solution



My goal is to find numbers of elements in intersections n(AB), n(AC), n(ABC) from the given data.


As soon as I determine these values, I will be in position to apply the Inclusion-Exclusion principle/formula.
It will give me answers to questions (B) and (C).


Step 1.  From (1) we know that n(A) = 65.  From (5) we know that n(A \ B) = 23.

         It means that n(A n B) = 65 - 23 = 42.   Again, n(AB) = 42.



Step 2.  From (1) we know that n(A) = 65.  From (6) we know that n(A \ C) = 12.

         It means that n(A n C) = 65 - 12 = 53.   Again, n(AC) = 53.



Step 3.  From step 2, we learned that n(AC) = 53.  From (6) we know that n(AC \ B) = 18.

         It means that n(ABC) = 53 - 18 = 35.



Step 4.  Let summarize what we know so far:  

         n(A) = 65 from (1);      n(B) = 140 from (2);     n(C) = 145 from (3);  

        n(AB) = 42 from step 1;   n(AC) = 53 from step 2;  n(BC) = 115 from (4);

        n(ABC) = 35 from step 3.


        Now I am in position to apply the Inclusion-Exclusion principle to find n(A U B U C).


        The formula of the  Inclusion-Exclusion principle is


            n(A U B U C) = n(A) + n(B) + n(C) - n(AB) - n(AC) - n(BC) + n(ABS) = substitute all the components, what we just know = 

                         =  65  + 140  + 145  -  42   -  53   - 115   +  35 = 175.


ANSWER to question B :  The number of housewives who use all 3 soaps is n(ABC) = 35.

ANSWER to question C :  The number of housewives who do not use any soaps of the three brands above is 

                        180 - n(A U B U C) = 180 - 175 = 5   (the complement to 180 of  n(A U B U C) = 175).