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In her extensive library, Christie has 42 fantasy books, 35 science fi

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In her extensive library, Christie has 42 fantasy books, 35 science fi  [#permalink]

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New post 21 Mar 2017, 05:14
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Question Stats:

56% (02:03) correct 44% (02:22) wrong based on 97 sessions

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In her extensive library, Christie has 42 fantasy books, 35 science fiction books, 32 historical fiction books, and 28 nonfiction books. If she removes books at random, how many must she remove to guarantee that she has removed at least 35 books of the same type?

A. 77
B. 105
C. 128
D. 129
E. 144

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Re: In her extensive library, Christie has 42 fantasy books, 35 science fi  [#permalink]

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New post 21 Mar 2017, 08:37
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Bunuel wrote:
In her extensive library, Christie has 42 fantasy books, 35 science fiction books, 32 historical fiction books, and 28 nonfiction books. If she removes books at random, how many must she remove to guarantee that she has removed at least 35 books of the same type?

A. 77
B. 105
C. 128
D. 129
E. 144


To answer this question, let's first determine the MAXIMUM number of books that can be removed WITHOUT having 35 books of the same type.

To start, we can remove all 32 historical fiction books, and 28 nonfiction books WITHOUT having 35 books of the same type.
From there, we can remove 34 fantasy books and 34 science fiction books WITHOUT having 35 books of the same type.
So far we have removed 128 books (32 + 28 + 34 + 34 = 128) and we do not have 35 books of the same type.

At this point, all that remains are 8 fantasy books, 1 science fiction book.
So, once we choose the next book, we will be GUARANTEED to have at least 35 books of the same type.
128 + 1 = 129
So, once we select 129 books, we will be GUARANTEED to have at least 35 books of the same type.

Answer: D

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Brent
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Re: In her extensive library, Christie has 42 fantasy books, 35 science fi  [#permalink]

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New post 21 Mar 2017, 10:06
Bunuel wrote:
In her extensive library, Christie has 42 fantasy books, 35 science fiction books, 32 historical fiction books, and 28 nonfiction books. If she removes books at random, how many must she remove to guarantee that she has removed at least 35 books of the same type?

A. 77
B. 105
C. 128
D. 129
E. 144


A good question. :D

    • To ensure that she removes 35 books of the same type, we must consider the worst-case scenario.
    • At present we have
      o Fantasy books - 42
      o Science Fiction - 35
      o Historical Fiction - 32
      o Non - fiction - 28
    • Thus, the worst case would be to take out books of each type, either equal to 34(since we need 35) or if the books are less than 34, then we take them all.
    • Therefore,
      o Fantasy books - 34
      o Science Fiction - 34
      o Historical Fiction - 32
      o Non - fiction - 28
      o Total books taken out so far = 34 + 34 + 32 + 28 = 128

    • Now observe, the even after we have taken 128 books, we still don't have 35 books of one type with us.
      o But now when we take out the 129th book, it is guaranteed that either it will the 35th Fantasy book or the 35th Science, ensuring that we get 35 books of one type.

Thus the correct answer is 129, Option D


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Re: In her extensive library, Christie has 42 fantasy books, 35 science fi  [#permalink]

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New post 06 Oct 2018, 07:52
First, note that Christie can only remove 35 or more books from fantasy or science fiction;

there aren’t enough historical fiction or nonfiction books, so we can remove those 60 books.

This leaves us with 77 books (42 fantasy and 35 science fiction).

Let’s imagine our worst possible outcome: she removes as many books of each type as possible without reaching a total of 35 from either type: 34 fantasy and 34 science fiction (a total of 68, bringing us to 128 total books removed).

Whichever book she picks next will give her 35 of either remaining type. If she removes a total of 129 books, she is guaranteed to have removed at least 35 science fiction or 35 fantasy books.
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Re: In her extensive library, Christie has 42 fantasy books, 35 science fi   [#permalink] 06 Oct 2018, 07:52
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