How many use at least one brand?

Question

In a survey of computer users, it was found that 50 use HP printers, 30 use IBM printers, 20 use Apple printers, 13 use HP and IBM, 9 use HP and Apple, 7 use IBM and Apple, and 3 use all three. How many use at least one of these Brands?

Answer 1

One way to solve this problem is to use a Venn Diagram.  Draw 3 intersecting circles and label them HP, IBM and Apple.  Start from the center region which is within all 3 circles and fill in the number 3, since 3 people use all 3.  Then move to the portion where only the HP and IBM circle intersect and fill in 10, since 13 use HP and IBM but this includes the 3 who use all 3 so there are 10 others who use only HP and IBM.  Similarly fill in the region where only HP and Apple intersect with (9-3)=6, and fill in the region where only IBM and Apple intersect with (7-3)=4.  Now fill in the number in the part of the HP circle that does not intersect either other circle.  We know there must be 50 total in the HP circle, and we have already filled in 3 (in all 3) plus 10 (in HP and IBM) plus 6 (in HP and Apple).  So there must be 50-(3+10+6)=50-19=31 in the only HP part of the circle.  Similarly there must be 30-(3+10+4)=30-17=13 in the only IBM section of the IBM circle, and 20-(3+6+4)=20-13=7 in the only Apple section of the Apple circle.  Now add up all the numbers you have written in the 7 different parts of the intersecting circles to get 31+10+13+6+3+4+7 = 74.  

 

Another way to do this problem is to use the principle of Inclusion-Exclusion.  Start by adding up (or including) all the people who use any 1 of the brands of printers (this is 50 HP + 30 IBM + 20 Apple = 100).  But we have double counted the people who use exactly 2 brands so we must subtract off (or exclude) all the people who use exactly 2 brands (this is 13 HP and IBM + 9 HP and Apple + 7 IBM and Apple = 29).  Finally, people who use all 3 brands have been included 3 times in the initial addition and excluded 3 times in the subsequent subtraction so we must add them in again to make sure they are included.  This gives 100 - 29 + 3 = 74 users total.