Problem Solving

VJesus12 wrote:In how many different ways can 3 fiction books and 3 non-fiction books be arranged in a row of 6 books on a shelf such that the fiction books are not separated, and the non-fiction books are not separated?

A. 24
B. 36
C. 72
D. 144
E. 288

[spoiler]OA=C[/spoiler]

Source: Magoosh

The 3 fiction books can be arranged in 3! = 6 ways. Similarly, the 3 nonfiction books can be arranged in 3! = 6 ways also. Since we can arrange the fiction books before the nonfiction books, or the nonfiction before the fiction books, the number of ways to arrange the 6 books is:

6 x 6 x 2 = 72

Answer: C

VJesus12 wrote:In how many different ways can 3 fiction books and 3 non-fiction books be arranged in a row of 6 books on a shelf such that the fiction books are not separated, and the non-fiction books are not separated?

A. 24
B. 36
C. 72
D. 144
E. 288

[spoiler]OA=C[/spoiler]

Source: Magoosh

Take the task of arranging the 6 books and break it into stages.

Stage 1: Arrange the 3 fiction books in a row
We can arrange n unique objects in n! ways
So, we can arrange the 3 books in 3! ways (= 6 ways)
So, we can complete stage 1 in 6 ways

Stage 2: Arrange the 3 non-fiction books in a row
We can complete stage 2 in 6 ways

Now that we've arranged the two types of books, we need to determine the order they appear on the shelf (i.e.. fiction-nonfiction or nonfiction-fiction)

Stage 3: Select the order in which the 2 book types appear on the shelf
There are 2 options: fiction-nonfiction or nonfiction-fiction
So, we can complete stage 3 in 2 ways

By the Fundamental Counting Principle (FCP), we can complete all 3 stages (and thus arrange all 6 books) in (6)(6)(2) ways (= 72 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 our free 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