Three Set Example Problems Problem 1 There are a total of 103 foreign language students in a high school where they offer Spanish French and German There are 29 students who take at least 2 languages at once If there are 40 Spanish students 42 French students and 46 German students how many students take all three languages at once We try to motivate this through the Venn diagram If you look at the counting Since all students are taking at least one language nothing is outside of the A or B or C set But we have an overlap since the count for A with B with C is 40 42 46 128 but we have only 103 students So the overlap is 128 103 25 only Now the statement that 29 students who take at least 2 languages at once includes the area shaded in ugly green So the common intersection red crosshatch is included in it redundantly It must be the source of disagreement between 25 and 29 and the overlap those taking all three is 29 25 4 We could also use the formula below The formula is developed by cutting apart the three set Venn in a way similar to the formula for two sets Venn diagrams in Lesson 3 1 Then by substitutions Arizona State University Department of Mathematics and Statistics 1 of 2 Three Set Example Problems Problem 2 In a survey of 279 college students it is found that 29 like both Brussels sprouts and broccoli 25 like both Brussels sprouts and cauliflower 24 like both broccoli and cauliflower and 14 of the students like all three vegetables 68 like brussels sprouts 90 like a broccoli 57 like cauliflower In a problem like this one a Venn Diagram is the best organizational tool This is what I call Mickey s Head in the videos and such Start with three interlocking circles Each circle represents the count of a characteristic IE A like brussels sprouts B like broccoli and C like cauliflower Label the circles in a way to remember what it stands for The next step is to find something you can fill in without any calculation Look for words like all or none In this problem 14 of the students like all That means the number 14 goes into the area of Mickey s nose Now we look at the pairwise intersections These are our and statements using only two characteristics In the second diagram they are shaded gray Recall that these overlap the all region The 14 we know about are already counted The three red results show how we adjust the counts so the cat s eye shapes total properly for these statements 29 like both Brussels sprouts and a broccoli 25 like both Brussels sprouts and cauliflower 24 like both broccoli and cauliflower The rest of the diagram works in a similar way Totals in each circle must add to meet these statements 68 like brussels sprouts 90 like a broccoli 57 like cauliflower So the A only region must have a count of 68 15 14 11 28 Then the B only region must have a count of 90 15 14 10 51 Finally the C only region must have a count of 57 11 14 10 22 We filled them into the diagram to see the third result We are almost done The possibility does exist that there is someone in the survey who doesn t like any of these three taste delights To fill in the likes none of them area add all the numbers in Mickey s head to get 151 Since the survey includes 279 students the number counted outside of the head is 279 151 128 That is shown in green The diagram is complete On to Lesson 6 2 of 2 Arizona State University Department of Mathematics and Statistics