In a survey of 270 college students, it is found that 64 like brussels sprouts, 94 like broccoli, 58 like cauliflower, 26 like both brussels sprouts and broccoli, 28 like both brus- sels sprouts and cauliflower, 22 like both broccoli and cauliflower, and 14 like all three vegetables. How many of the 270 students do not like any of these vegetables?
I don't have any idea for solving this
can anyone help?
thanks
On
You have three sets - the set $S$ containing people who like brussel sprouts, the set $B$ of people who like broccoli and the set $C$ of people who like cauliflower. You know $|S|$, $|B|$, $|C|$, $|S\cap B|$, $S\cap C|$, $|B\cap C|$ and $|S\cap C\cap B|$. You're looking for $|S\cup C \cup B|$. To you know any formula that relates those quantities? Does inclusion-exclusion principle ring a bell?
I have drawn a Venn diagram and started filling it out according to the information given. Here it is (you don't need to write letters in your diagram, they're just for me to refer back to, so I can say "Area $C$" instead of "The area that is inside the light-green and the dark-green circles, but outside the teal circle"):
The way these things work, is that a student is placed inside one of the circles if he / she likes the corresponding vegetable, and outside that circle if he / she doesn't like it. Then we write a number in each zone corresponding to how many students have been placed there. So for instance, the area marked $A$ is where we would put all the students who like broccoli and cauliflower, but dislike Brussel sprouts (note that you have not been told yet how many there are of these students, you only know how many like broccoli and cauliflower, regardless of their stance on Brussels sprouts).
Now, we have been given some information about the number of students inside each circle. For instance, we've been told that there are $14$ students who like all three, so I've gone ahead and placed that in the center. Now we need to work our way outward. We've been told that there are $22$ students who like broccoli and cauliflower. That means that there are $22$ students in the wedge between both circles. $14$ of them are in the part inside the Brussels sprout circle, so the remaining $8$ have to be outside it. Thus where there is an $A$ there are $8$ students. (Now would be a good time to write an $8$ in that area on your own Venn diagram, which you have made, have you not?)
The $B$ area is for those who like cauliflower and Brussels sprouts, but dislike broccoli. We've been told that there are $28$ in total who like Brussels sprouts and cauliflower, and we've already placed $14$ of them in the center zone, so that means there are 14 left to place in the $B$-zone. I am confident you can figure out the $C$-zone on your own.
Now we get to the last three "inner" zones, reserved for those who like one specific vegetable, but dislike the two others. Starting with $D$: We are told that there are $58$ students in total contained in the cauliflower circle. We've already placed $8 + 14 + 14 = 36$ of them (area $A$, area $B$ and the center zone), so that means that there are $58 - 36 = 22$ in area $D$.
$E$ and $F$ are done the same way (but with their corresponding numbers, so the actual numbers you get might not end up the same).
Finally you get to the answer: You're interested in how many students we have left to place (that is what the question asks for, after all). So what you do is, you add up all the students you've placed so far, and then you see how many you have left to place. Those have to go in the $G$-zone. And there you have it.