Question 1113348
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I was curious to see what methods of solution other tutors might come up with for this problem....<br>
There is a flaw in the analysis by the other tutor, leading to incorrect answers.<br>
The easiest way to solve this problem is by using the inclusion-exclusion principle.  That principle, with 3 brands, says that the total number of people sampled is equal to
(sum of numbers who liked one brand) minus (sum of numbers who liked two brands) plus (number who liked all three brands).<br>
Note that in order to solve the problem we have to make the assumption (not stated in the problem) that every one of the people liked at least one of the brands.<br>
So if x is the number of people who liked all three brands, then
{{{(250+250+283)-(50+55+65)+x = 625}}}
{{{783-170+x = 625}}}
{{{613+x = 625}}}
{{{x = 12}}}<br>
So 12 people liked all three brands.  Then, using that number with the given information, we can determine how many people liked which brands:<br>
PQR: 12
PQ:  50-12 = 38
PR:  65-12 = 53
QR:  55-12 = 43
P:   250-38-53-12 = 147
Q:   250-38-43-12 = 157
R:   283-53-43-12 = 175<br>
And now we can answer the specific questions that were asked.<br>
(i) all three brands: 12
(ii) exactly one brand: 147+157+175 = 479
(iii) at least two brands: 12+38+53+43 = 146
(iv) P and Q but not R: 38<br>
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<br>
The inclusion-exclusion principle gets you to the answer quickly.  But you can also find the number of people who like all three brands -- and thus finish the problem -- by using a Venn diagram with three circles representing brands P, Q, and R.<br>
If x is again the number of people who like all three brands, then<br>
PQR = x<br>
PQ = 50-x
PR = 65-x
QR = 55-x<br>
P = 250-(50-x)-(65-x)-x = 135+x
Q = 250-(50-x)-(55-x)-x = 145+x
R = 283-(65-x)-(55-x)-x = 163+x<br>
The sum of all those numbers has to be the total number of people:
{{{x+(50-x)+(65-x)+(55-x)+(135+x)+(145+x)+(163+x) = 625}}}
{{{x + 613 = 625}}}
{{{x = 12}}}<br>
And from there you finish the problem as earlier.