Probability and Venn Diagrams The question in concern asks, "$136$ people were asked if they liked Math, Science, or Social Studies. They all liked at least one. 
$40$ like Math
$17$ like Math and Science
$40$ like Science
$8$ like Science and Social Studies
$32$ like Social Studies
$11$ like Math and Social Studies
$6$ like all three subjects."
My question is that although it says $136$ people were surveyed, the results that were gathered afterwards add up to $154$. Where are the extra $18$ people coming from exactly?
Thanks in advance.
 A: If I were given these numbers in real life I would, like you, interpret the claim that '$40$ like Math' as '40 like Math but not Science and also not social science', given that it has separate numbers for all kinds of combination of liking subject areas. So, I can understand your confusion!
But, given as that does indeed add up to more than $136$, probably the claim means that '40 like math  .... and possibly science or social science as well'
Thus, for example, the $17$ people that like Math and Science are part of the $40$ people that like Math.
As such, you can actually use a Venn diagram to figure out exactly how many people are in each of the different regions of that diagram. For example, given that $6$ people like all three subjects, and given that $17$ people like both Math and Science, we can infer that $17-6=11$ people like Math and Science but not Social Science.
In fact, given that there will be quite a bit of overlap between the different subject areas, you'll probably get quite a few people less than $136$ that likes any subject at all: probably quite a few don't like any of the three subjects! Again, draw a Venn diagram, and you can work out all the numbers.
A: What you were probably missing is that one person can like more than one subject.
Let me give you a more simple example, with $2$ subjects.

$10$ people like math, $20$ like science, and $8$ like both.

First, jump to $8$ like both.
There are $10-8=2$ people who like math only.
There are $20-8=12$ people who like science only.
Now, add $8$ into your counting - the $8$ people who like both math and science.
So, there are $2+12+8=22$ people in total.
Implement this same logic, this time for $3$ subjects, and start by drawing a Venn Diagram!
Hint: Start from in to out (start from excluding people who like all $3$ subjects and counting them all the way at the end).
A: It is a question on Venn diagrams and the data provided contains intersections. Clearly the intent is that the sets, M,S,T (for Maths, Science, and Social Studies):
$$\begin{split}n(\Omega)&=136\\n(M)&=40\\ n(M\cap S)&=17\\n(S)&=40\\n(S\cap T)&=8\\ n(T)&=32\\n(M\cap T)&=11\\ n(M\cap S\cap T)&=6\end{split}$$
Now, were you told that everbody liked at least one, or were you asked to find how many liked at least one?  Because the data would seem to state, by the principle of inclusion and exclusion, that
$$n(M\cup S\cup T)~{=n(M)+n(S)+n(T)-n(M\cap S)-n(M\cap T)-n(S\cap T)+n(M\cap S\cap T)\\\ll 136}$$
