Problem Solving

VJesus12 wrote: ↑

Sat May 30, 2020 3:09 pm

A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible?

(A) \(\dfrac{10!}{4!}\)
(B) \(4!\cdot 6!\)
(C) \(4!\cdot 7!\)
(D) \(4!\cdot 10!\)
(E) \(4!\cdot 6!\cdot 10!\)

[spoiler]OA=C[/spoiler]

Source: Magoosh

Take the task of arranging the 10 books and break it into stages.

Stage 1: Arrange the 4 books about Abe Lincoln in a row
We can arrange n objects in n! ways.
So, we can arrange the 4 books in 4! ways

IMPORTANT: Now we'll "glue" the 4 Abe Lincoln books together to form 1 SUPER BOOK (this will ensure that the 4 Abe Lincoln books remain together)
So, we now have 1 Abe Lincoln SUPER BOOK, along with 6 non-Abe Lincoln books (for a total of 7 "books")

Stage 2: Arrange the 7 "books"
We can arrange n objects in n! ways.
So, we can arrange the 7 books in 7! ways

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus arrange all of the books) in (4!)(7!) ways

Answer: C
--------------------------

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. For more information about the FCP, watch this video: https://www.gmatprepnow.com/module/gmat- ... /video/775

You can also watch a demonstration of the FCP in action: https://www.gmatprepnow.com/module/gmat ... /video/776

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/what-should- ... 67256.html
- https://www.beatthegmat.com/counting-pro ... 44302.html
- https://www.beatthegmat.com/picking-a-5- ... 73110.html
- https://www.beatthegmat.com/permutation- ... 57412.html
- https://www.beatthegmat.com/simple-one-t270061.html


MEDIUM
- https://www.beatthegmat.com/combinatoric ... 73194.html
- https://www.beatthegmat.com/arabian-hors ... 50703.html
- https://www.beatthegmat.com/sub-sets-pro ... 73337.html
- https://www.beatthegmat.com/combinatoric ... 73180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-sep ... 71047.html
- https://www.beatthegmat.com/combinatoric ... 67079.html


DIFFICULT
- https://www.beatthegmat.com/wonderful-p- ... 71001.html
- https://www.beatthegmat.com/permutation- ... 73915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/no-two-ladie ... 75661.html
- https://www.beatthegmat.com/combinations-t123249.html


Cheers,
Brent

VJesus12 wrote: ↑

Sat May 30, 2020 3:09 pm

A librarian has a set of ten books, including four different books about Abraham Lincoln. The librarian wants to put the ten books on a shelf with the four Lincoln books next to each other, somewhere on the shelf among the other six books. How many different arrangements of the ten books are possible?

(A) \(\dfrac{10!}{4!}\)
(B) \(4!\cdot 6!\)
(C) \(4!\cdot 7!\)
(D) \(4!\cdot 10!\)
(E) \(4!\cdot 6!\cdot 10!\)

[spoiler]OA=C[/spoiler]

Solution:

Since the 4 Lincoln books must be together, we can, for now, treat them as just 1 book. Since there are 6 other books, there are a total of 1 + 6 = 7 books, and hence there are 7! arrangements. However, within the 4 Lincoln books, there are 4! ways to arrange them. Therefore, the total number of arrangements of all 10 books, with the 4 Lincoln books together, is:

4! * 7!

Answer: C