Lesson Upper league combinatorics problem

Upper league combinatorics problem

Problem 1

We are given a universal set of 800 consumers.

We also are given its three basic subsets

    - C of 230 persons who took coffee;
    - T of 245 persons who took tea;
    - A of 325 persons who took cocoa.


We also are given subsets

    - CA of 70 persons who took coffee and cocoa (intersection of sets C and A);
    - Co of 110 persons who took coffee only  (notice suffix o = "only" at Co);
    - Ao of 185 persons who took cocoa only   (notice suffix o = "only" at Ao).


We also are given the triple intersection CTA of 30 persons who took all the three beverages.


    +-----------------------------------------------+
    |         Now I will make focus-pocus           |
    |    by extracting the numbers from nowhere.    |
    +-----------------------------------------------+


Since we know C and Co, we can find the number of people in the set (C\Co) by subtracting
230 - 110 = 120. The set (C\Co) is the set of people who took coffee and something else,
so (C\Co) is, actually, (CT U CA).  Thus n(CT U CA) = 120.


From (CT U CA), we can subtract CA (70 persons), and we will get then

    n(CT) = n(CT U CA) - n(CA) + n(CTA) = 120 - 70 + 30 = 80.


     +----------------------------------------------------+
     |  Next, we can make similar reasoning for A and Ao. |
     +----------------------------------------------------+


Since we know A and Ao, we can find the number of people in the set (A\Ao) by subtracting
325 - 185 = 140. The set (A\Ao) is the set of people who took cocoa and something else,
so (A\Ao) is, actually, (AT U CA).  Thus n(AT U CA) = 140.


From (AT U CA), we can subtract CA (70 persons), and we will get then

    n(AT) = n(AT U CA) - n(CA) + n(CTA) = 140 - 70 + 30 = 100.


So far, we learned (we can add it to the given information) that subsets CT and AT contain

    n(CT) = 80 persons;  n(AT) = 100 persons.


      +---------------------------------------------------------------+
      |   Now I am in position to answer questions (a), (b) and (c).  |
      +---------------------------------------------------------------+


(a)  Find the number of consumers who took coffee and tea only.

     CT_only = CTo = from CT subtract CTA = n(CT) - n(CTA) = 80 - 30 = 50.


(b)  Find the number of consumers who took tea and cocoa only.

     AT_only = ATo = from AT subtract CTA = n(AT) - n(CTA) = 100 - 30 = 70.


(c)  Find the number of consumers who took tea only.

     T_only = To = from T subtract CT and AT and add CTA = 
                 = n(T) - n(CT) - n(AT) + n(CTA) = 245 - 80 - 100 + 30 = 95.


The problem is just solved.


ANSWER.  (a) The number of consumers who took coffee and tea only is 50.
         (b) The number of consumers who took tea and cocoa only is 70.
         (c) The number of consumers who took tea only is 95.