Problem Solving

. Of the science books in a certain supply room, 50 are on botany, 65 are on zoology, 90 are on physics. 50 are on geology, and 110 are on chemistry. If science books are removed randomly from the supply room, how many must be removed to ensure that 80 of the books removed are on the same science?
(A) 81
(B) 159
(C) 166
(D) 285
(E) 324

OA is E . Can any one throw some light on this?

sairamGmat wrote:. Of the science books in a certain supply room, 50 are on botany, 65 are on zoology, 90 are on physics. 50 are on geology, and 110 are on chemistry. If science books are removed randomly from the supply room, how many must be removed to ensure that 80 of the books removed are on the same science?
(A) 81
(B) 159
(C) 166
(D) 285
(E) 324

OA is E . Can any one throw some light on this?

The sciences videlicet botany, zoology, and geology do not have enough books to ensure that 80 of the books removed are on the same science. Physics 90 and chemistry 110 are the true contenders left.

Hence, if all books of the sciences videlicet botany, zoology, and geology were removed from the supply room, and then we can still say that we are not sure even after removing 165 books, that 80 of the books removed are on the same science. We still need to remove minimum of the remaining 200 books to meet the said condition.

But, how many more books after removing 165 books would allow us meet the said condition?

Well, we can contentedly eliminate choices A through C now.

D is 120 in excess to 165. Just think, does the removal of 120 meet the said condition, if yes, it's D, otherwise E.

Think of the worst case with D, it's possible that 60 books of each science were have been removed so far. Now go for E without checking it.

Why E?

E is 159 in excess to 165. Just think, does the removal of 159 meet the said condition? The worst case could be when 79 books of each science were have been removed, that's a total of 158 books were have been removed and we are still not sure, just 1 more book of any of the two sciences, and we can bet.

Hey guys,

Nice use of the answer choices by Sanju - don't let me step on your toes, but I figured I'd chime in with an explanation of how to just purely solve this, too, since I've seen a few variations of "how many are necessary to guarantee/ensure that X condition is met" questions.

In order to guarantee that we get 80 of the same science, we have to look at the "worst case scenario" of continuing to draw books and never finding an 80th match until it's absolutely certain.

Well, that means that we draw the maximum allowable number of each books until the last book must be an 80th copy. So, we would end up with 79 of each book for which there are even 79 possible books, and the maximum number of all of the other books, until the last draw is bound to be the 80th copy of one.

We have 50 botany, 65 zoology, and 50 geology - none of those is even a possibility to hit 80, so our "worst case scenario" should include all of those, or a total of 165 books.

Then, we'd want to have the max number of each of physics and chemistry before getting to 80, so we'd have 79 of each of those, or a total of 158 books.

At this point, with 165 + 158 = 323 books without an 80th copy, the last draw is guaranteed to get us our 80th match, and so we add one for that last draw to get to 324 books.


To summarize, if the question asks you to guarantee a certain outcome, you can calculate the maximum number of outcomes before that final one is met - just max out all of the possibilities until you have no choice but to get what they ask for.