In a survey of 63 people, 23 people subscribe to magazine A, 21people subscribed to magazine B,and 17 people subscribed to magazine C. For any two of the magazines,4 people subscribed to both magazines but not to the third magazine. If 5 people in the survey did not subscibe to any of the three magazines,how many people subscribed to all three magazines?

My sister told me the answer is 9 but I still don't know how to do it. Thanks

BTW, the choices are

a)9
b)7
c)5
d)2
e)It cannot be determined from the info given.

best done with Venn diagram

draw 3 intersecting circles, place x in the intersection of all three since we are not given that number

look at the intersection of any two circles.
We already have x in the middle, so place 4 in each of the remaining region of the pairs

now go to each of the whole circles
in A we already have accounted for 8+x, so the open part of A is 23-8-x or 15-x
in the same way the open part of B is 13-x
and the open part of C is 9-x

now add them all up
15-x + 13-x + 9-x + 12 + x + 5 = 63
-2x = -9
x = 4.5

not possible, unless I made an arithmetic error,
there is something wrong with your data

What do you mean by the open part. Thanks

the part of the circle not covered by any part of the other two.

And shouldn't the answer be x=-4.5; I think you had make a mistake, but thanks

it should be 36 people in the survey not 63. I believe there was a typing error on this question.

To solve this problem, we can use a method called the Principle of Inclusion-Exclusion.

First, let's start by finding the total number of people who have subscribed to at least one magazine. We can do this by adding up the number of subscribers for each magazine:

Number of people subscribed to at least one magazine = Number of people who subscribed to magazine A + Number of people who subscribed to magazine B + Number of people who subscribed to magazine C

= 23 + 21 + 17

= 61

Now, we need to take into account the fact that some people have subscribed to more than one magazine. We are given that 4 people subscribed to any two magazines but not to the third magazine. Let's say these 4 people can be divided into three groups: AB, BC, and AC, representing the people who subscribed to magazine A and magazine B but not magazine C, the people who subscribed to magazine B and magazine C but not magazine A, and the people who subscribed to magazine A and magazine C but not magazine B, respectively.

Since the same 4 people are counted in both the AB and AC groups (as they have subscribed to both magazines A and B), we need to subtract the number of people in these groups to avoid double counting. Therefore, we subtract 4 from the total number of people who have subscribed to at least one magazine:

Adjusted total = Total number of people subscribed to at least one magazine - Number of people who subscribed to AB group - Number of people who subscribed to BC group - Number of people who subscribed to AC group

= 61 - 4 - 4 - 4

= 49

Now, we need to consider the 5 people who did not subscribe to any of the three magazines. We need to subtract these 5 people from the adjusted total:

Number of people subscribed to at least one magazine but not all three = Adjusted total - Number of people who did not subscribe to any magazine

= 49 - 5

= 44

Since we are asked to find the number of people who subscribed to all three magazines, we can subtract this result from the adjusted total:

Number of people who subscribed to all three magazines = Adjusted total - Number of people subscribed to at least one magazine but not all three

= 49 - 44

= 5

Therefore, the correct answer is that 5 people subscribed to all three magazines, not 9.