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There are 8 books on a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at random

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by Vincen » Thu Apr 15, 2021 11:16 am

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There are 8 books on a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at random without replacement, how many different combinations are there that at least one paperback book is selected?

A. 40
B. 45
C. 50
D. 55
E. 60

Answer: D

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Re: There are 8 books on a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at rand

by Brent@GMATPrepNow » Sat Nov 20, 2021 6:21 am
Vincen wrote: ↑
Thu Apr 15, 2021 11:16 am
 There are 8 books on a shelf that consist of 2 paperback books and 6 hardback books. If 4 books are selected at random without replacement, how many different combinations are there that at least one paperback book is selected?

A. 40
B. 45
C. 50
D. 55
E. 60

Answer: D

Source: GMAT Prep
In general, when it comes to counting questions involving "at least," we can use the following property:
Number of outcomes that satisfy the restriction = (number of outcomes that ignore the restriction) - (number of outcomes that BREAK the restriction)

So we get:
Number of combinations with at least one paperback = (total number of 4-book combinations) - (number of 4-book combinations that DON'T have at least one paperback)

In other words:
Number of combinations with at least one paperback = (total number of 4-book combinations) - (number of 4-book combinations with ZERO paperbacks)

total number of 4-book combinations
There are 8 altogether, and we must choose 4 of them
Since the order in which we select the books does not matter, we can use combinations.
We can select 4 books from 8 books in 8C4 ways (= 70)

number of 4-book combinations with ZERO paperbacks
In order to get ZERO paperbacks, we'll select 4 books from the 6 hardback books only.
We can select 4 books from 6 books in 6C4 ways (= 15)

So, the number of combinations with at least one paperback = (70) - (15) = 55

Answer: D
Brent Hanneson - Creator of GMATPrepNow.com
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