Of the science books in a certain supply room, 50 are on botany, 65 ar

We can remove 80 books only from physics or from chemistry. Worst case scenario would be if we remove all books on botany, zoology, geology, 79 books on physics and 79 books on chemistry.

We would have 50(botany) + 65(zoology) + 50(geology) + 79(physics) + 79(chemistry) = 323 books and still won't have 80 books from the same science. As after this, only books on physics and chemistry will be left then any next book would become the 80th book either on physics or on chemistry.

Answer: E.

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Hope it helps.
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What is the number at which picking one more book will guarantee that a thresold of 80 has been surpassed in one science?

Imagine you are playing a rigged game. The object of the game is for you to draw 80 books from the same science. The person attempting to cheat you out of winning picks out the books for you. He or she tries to prevent you from drawing the winning number of books until it is no longer possible to do so. He or she first starts off by picking books that cannot reach the 80 threshold and does so until such books are exhausted. He or she then starts picking books out of one the two remaining sciences until he reaches 79. To delay reaching the 80 thresold as much as possible he or she then shifts to the last science until he or she reaches 79. At this point it is no longer possible to cheat you out of winning and when the next book is drawn, you meet the 80 book threshold. The number of books drawn at that point is therefore 50 + 65 + 50 + 79 + 79 + 1 = 324.

Thanks Bunel. Very helpful response yet again!

Is this the only way to solve this?

Pretty much yes. For such kind of questions you should test worst case scenario to get the answer. Check the links in my previous post for similar questions.
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Check other Worst Case Scenario Questions from our Special Questions Directory.
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I had a different approach.

You know that only books you can remove to get 80 of the same science is Physics (90) or Chemistry (110). Probability of drawing out a Chemistry is higher with every draw 110/365 vs 80/365. Logically you'll need to take out fewer books to 80 of the Chemistry. So you look at Physics books instead. Because the question says 'ensure'.

P(Physics) = 90/365 simplify to 90/360 or 1/4. You know that for every 4 books you take out 1 is likely to be a physics book. Then to get 80 physics books you need to take out 80 x 4 = 320. Answer is (E)

Had an alternative approach

To ensure you get at least 80 books, you only look at Physics or Chemistry. 90 or 110. I calculated what percentage 80 books were of each group and took the highest percentage (i.e. 80/90 for Physics) and multiplied it by the total number of books (365). This makes sure you get 80 of one science as it makes sure you get 80 of the science that has the fewer books (i.e. harder probability). This gives you exactly 324.

Let’s assume the worst case scenario: all 50 botany, all 50 geology, and all 65 zoology books are removed, and 79 physics and 79 chemistry books are removed. We see that we have 50 x 2 + 65 + 79 x 2 = 323 books removed, but we still don’t have 80 books of one science removed. However, if we remove one more book, which must be either a physics book or a chemistry book, we will have 80 books of one science removed. Thus we need need to remove 323 + 1 = 324 books.

Answer: E
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Let's see what happens when the following books are removed:
50 on botany
65 on zoology
79 on physics
50 on geology
79 on chemistry
TOTAL = 323

At this point, we do NOT have 80 of any one science topic.
However, if we remove one of the remaining books (which are either physics books or chemistry books), we will certainly have 80 books on one science topic.
In other words, if we remove 324 books, we can be certain of having 80 books on one science topic

Answer: E

Cheers,
Brent

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