Hi. I am super confused about this question. I thought |at least 1| = |A| + |B| + |C| - |exactly 2| + |exactly 3|. So I did 150 = 60+75+80+10-|exactly 2| and solved for |exactly 2|, yielding 75 with exactly 2. But then all of a sudden it’s telling me to multiply |exactly 3| by 3 for the correct answer in the formula, which seems like we are ignoring the original formula and just manipulating my answer into the correct answer without grounding. Please help! Thank you so much!
Try attempting this Question using Venn Diagrams instead of the formula.
Thanks Jovyn. For some reason I just have a really hard time visualizing the overlaps, especially with more than 2 sets. So an explanation in terms of the formula would be really be helpful.
The multiplying by 3 is because we need to account for all the 3 types of cars.
Hey hi
I think the formula you are using is not completely correct.
Correct formula:
|A|+|B|+|C| - |A∩B|-|A∩C|-|B∩C|+|A∩B∩C|.
This can look similar to what you were using but the middle part is not really “exactly 2”. ‘A∩B’ is presence of both A and B but also includes both presence of C and absence of C, i.e. (A and B with C) + (A and B without C).
Now on solving this by plugging the rest we get:
|A∩B|+|A∩B|+|B∩C| = 75 (which is what you were able to compute as well).
This is not the final answer because we need exactly “2” so we need to remove the count of all the three being present:
Only A and B present = |A∩B| - |A∩B∩C|
Only B and C present = |B∩C| - |A∩B∩C|
Only A and C present = |A∩C| - |A∩B∩C|
Adding all these to get exactly “2” being present = |A∩B|+|A∩B|+|B∩C| - 3|A∩B∩C| = 75 - 30 = 45.
Hope this helps!