This is a worst case scenario question.

Consider you're the unluckiest person in the world: then when you take some books out of the library, it just so happens that they're all from the sets that DON'T have at least 80 books, which means that you take out 50 + 50 + 65 = 165 books with no success. Then you start on the other two sets. You might think that taking an extra 80 books out will suffice, but what if you take 40 out of the physics books and 40 out of the chemistry books? To be 100% sure you've taken at least 80 books from one subject, you need to take out another 80 + 79 = 159 books. This way, you're absolutely sure you've gotten the required quota out of there.

Take less than 159 books out of there, and you might be carrying 79 books of one subject and 79 books of the other. But take the 159 and you're 100% sure you've taken at least 80 books of the same subject.

The grand total will be 165 + 159 = 324.

DanaJ wrote:. Then you start on the other two sets. You might think that taking an extra 80 books out will suffice, but what if you take 40 out of the physics books and 40 out of the chemistry books? To be 100% sure you've taken at least 80 books from one subject, you need to take out another 80 + 79 = 159 books. This way, you're absolutely sure you've gotten the required quota out of there.

Take less than 159 books out of there, and you might be carrying 79 books of one subject and 79 books of the other. But take the 159 and you're 100% sure you've taken at least 80 books of the same subject.

The grand total will be 165 + 159 = 324.

I did'nt understand the last part.Where did the 79 come from??

This is more of a reasoning question than maths..
we want to know how many books MUST be removed to ensure that 80 of the books removed are on the same science. so we have to take the worst case.
Now botany,zoology and geology books are less than 80 in number. we are looking for worst case, so let's say we are running a bad luck and we get them first
.so 50+50+50
now lets say we got 79 physics books and our luck is so bad that we don't get the 80th one(worst case,remember?). thats why we add 79 here. same for the chemistry books
after this whatever book we get, it will complete 80 books of the same color condition
so 1 here
total=50+50+65+79+79+1
hope i helped