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Four newspapers - Hindu, ET, TOI and IE - operate in a town. No household that subscribed for ET also subscribed for IE. Further:
$(1)$ Number of households that subscribed for Hindu is $123$
$(2)$ Number of households that subscribed for Hindu and ET is $45$
$(3)$ Number of households that subscribed for Hindu and TOI is $46$
$(4)$ Number of households that subscribed for Hindu and IE is $43$
$(5)$ Number of households that subscribed for Hindu, ET and TOI is $18$
$(6)$ Number of households that subscribed for Hindu, TOI and IE is $22$
$(7)$ Number of households that subscribed for TOI is $107$
$(8)$ Number of households that subscribed for TOI and ET is $36$
$(9)$ Number of households that subscribed for TOI and IE is $46$
$(10)$ Number of households that subscribed for ET is $97$
$(11)$ Number of households that subscribed for IE is $91$

Question : How many households subscribed for at least two newspapers?

I prepared a $4$ set Venn diagram by putting in the data like this but it gave me wrong answers. What have I done wrong with my Venn diagram? enter image description here

Please help !!!

Thanks in advance !!!

Ganit
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  • The diagram look OK to me. What answers did you get wrong? – Vasili Jul 03 '22 at 11:46
  • +1 to your posting, for very nice presentation. I am going to duck the actual question that you posed, because in my opinion, use of a Venn diagram is not the best way to attack this problem. Instead, I suggest creating a truth table. As an example, see this answer. Note that if you decide to post work involving the truth table, you can pretend to edit my post in order to copy paste the MathJax associated with creating the chart. – user2661923 Jul 03 '22 at 11:46
  • @Vasili : I have updated the description of my problem. I have added the question asked. I got my answer for that question to be 256 but the correct answer is 136. I just added all the values that are present in my diagram. – Ganit Jul 03 '22 at 11:52
  • @user2661923 : I read the answer posted for another problem and I am wondering in my case as there are 4 cases, won't there be more cases for which I'll have to make the truth table? – Ganit Jul 03 '22 at 11:53
  • Absolutely. You will need $2^4$ rows, instead of $2^3$ rows. Despite this, I still advise the truth table over the Venn diagram. As the number of truth table variables increases (e.g. from $8$ to $16$), the Venn diagram complexity similarly increases. – user2661923 Jul 03 '22 at 11:55
  • I think you are double-counting those who signed up for three newspapers if you are just adding all the numbers. – Vasili Jul 03 '22 at 12:02
  • @user2661923 : Can you please tell me why this Venn diagram setup won't work in this scenario? Is it because there is nothing common between ET and IE group? – Ganit Jul 03 '22 at 12:05
  • I didn't look that closely at your specific posting. I have no opinion whether a Venn diagram approach to your problem is feasible. My personal bias (which others may well disagree with) is that it is easier to visualize the consequences of the premises, whatever the premises may be, if you simply assign a separate variable to each row of the truth table. – user2661923 Jul 03 '22 at 13:30
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    Draw 3 intersecting circles for ET, H and TOI, then draw another circle for IE which overlaps only H and TOI. It is then quite easy to fill in and arrive at the required answer - it took only a few minutes… – David Quinn Jul 03 '22 at 13:37
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    You've done (1) right, but not (2). Note that 45 = H and ET is not one square but the total of 4 squares making a 2x2 grid on top left of your diagram. Similarly for all other cases of two newspapers, your representation is incorrect, as A and B does not mean A and B only. – Macavity Jul 03 '22 at 16:04
  • @DavidQuinn : I used your approach and it worked for me. I got the correct answers. However won't the above 4 set Venn diagram work if I correct my mistakes in that Venn diagram as per the comment below yours? – Ganit Jul 04 '22 at 10:55
  • @Ganit yes I expect it will work but I haven’t tried it your way… – David Quinn Jul 04 '22 at 14:20

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