Question: 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) (10!)/(4!)
(B) (4!)(6!)
(C) (4!)(7!)
(D) (4!)(10!)
(E) (4!)(6!)(10!)
“A librarian has a set of ten books, including four different books about Abraham Lincoln” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "Official Guide for GMAT Review". GMAT Quant section consists of a total of 31 questions. To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1
Given in the question that 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.
It is asked in the question to find out the number of arrangements of the ten books is possible.
This is a question from permutation and combination.
In order to solve the problem, the candidate should remember the following formulas for permutation and combination.
Permutation of n objects taken r at a time: \(^{n}P{_r} = n! / (n-r)!\)
Combination of n objects taken r at a time: \(^{n}C_{r} = n! / ( (n-r)! * r!)\)
where x! = x*(x-1)*(x-2)*......(3).(2).(1)
Firstly there are four identical books of lincoln.
If we group them together and consider them as one then we’ll have 6 regular books and 1 group of lincoln books.
Now there are 7 books on the shelf and they can be arranged in 7! Ways.
Now remember that those 4 books that we consider as 1 are actually four books
So they can be arranged in 4! Ways among themselves.
Therefore the total number of combinations possible is - 7! * 4!
Option C is the correct answer.
Approach Solution 2
Let us first attach the 4 Lincoln books together to create one super book (this will ensure that the 4 books remain together)
We now have 7 books: 6 regular books and 1 super book
We can arrange these 7 books in 7! ways.
KEY: For each of the 7! arrangements, we can take the 4 Lincoln books (that comprise the SUPER BOOK) and arrange them in 4! ways.
So, the TOTAL number of arrangements = (7!)(4!)
Hence, the correct answer is C.
Approach Solution 3
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
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!
Hence, C is the correct answer.
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