• NEW! FREE Beat The GMAT Quizzes
Hundreds of Questions Highly Detailed Reporting Expert Explanations
• 7 CATs FREE!
If you earn 100 Forum Points

Engage in the Beat The GMAT forums to earn
100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Five letters A, P, P, L and E are listed in a row. How many tagged by: Max@Math Revolution ##### This topic has 4 expert replies and 0 member replies ### GMAT/MBA Expert ## Five letters A, P, P, L and E are listed in a row. How many ## Timer 00:00 ## Your Answer A B C D E ## Global Stats Difficult [Math Revolution GMAT math practice question] Five letters A, P, P, L and E are listed in a row. How many arrangements have at least one letter between the two Ps? A. 24 B. 30 C. 36 D. 42 E. 48 _________________ Math Revolution Finish GMAT Quant Section with 10 minutes to spare. The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. Only$149 for 3 month Online Course
Free Resources-30 day online access & Diagnostic Test
Email to : info@mathrevolution.com

### GMAT/MBA Expert

GMAT Instructor
Joined
09 Oct 2010
Posted:
1449 messages
Followed by:
32 members
59
Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

Five letters A, P, P, L and E are listed in a row. How many arrangements have at least one letter between the two Ps?

A. 24
B. 30
C. 36
D. 42
E. 48
$?\,\,\,:\,\,\,\# \,\,P{\text{s}}\,\,{\text{not}}\,\,{\text{together}}$
Total number of permutations of 5 elements, with exactly one repeating element appearing twice:
$\frac{{5!}}{{2!}} = 5 \cdot 4 \cdot 3$
Total number of permutations of 3 "single elements" (A,L,E) and one "double element" (PP):
$4!$
(We took into account the two P´s are indistinguishable, of course!)

$? = 5 \cdot 4 \cdot 3 - 4 \cdot 3 \cdot 2 = 12\left( {5 - 2} \right) = 36$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

### GMAT/MBA Expert

GMAT Instructor
Joined
25 May 2010
Posted:
15314 messages
Followed by:
1863 members
13060
GMAT Score:
790
Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

Five letters A, P, P, L and E are listed in a row. How many arrangements have at least one letter between the two Ps?

A. 24
B. 30
C. 36
D. 42
E. 48
Good arrangements = total arrangements - bad arrangements.

Total arrangements:
Number of ways to arrange 5 elements = 5!.
But when an arrangement includes IDENTICAL elements, we must divide by the number of ways each set of identical elements can be ARRANGED.
The reason:
When the identical elements swap positions, the arrangement doesn't change.
Here, we must divide by 2! to account for the two identical P's:
5!/2! = 60.

Let [PP] represent the 2 adjacent P's.
Number of ways to arrange the 4 elements [PP], D, G and T = 4! = 24.

Good arrangements:
Total arrangements - bad arrangements = 60-24 = 36.

_________________
Mitch Hunt
Private Tutor for the GMAT and GRE
GMATGuruNY@gmail.com

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

Available for tutoring in NYC and long-distance.
Student Review #1
Student Review #2
Student Review #3

Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! Sign up now.

### GMAT/MBA Expert

GMAT Instructor
Joined
08 Dec 2008
Posted:
12924 messages
Followed by:
1248 members
5254
GMAT Score:
770
Max@Math Revolution wrote:
[Math Revolution GMAT math practice question]

Five letters A, P, P, L and E are listed in a row. How many arrangements have at least one letter between the two Ps?

A. 24
B. 30
C. 36
D. 42
E. 48
Here's an approach that doesn't require us to subtract the bad arrangements.

Take the task of arranging the 5 letters and break it into stages.

Stage 1: Arrange the letters A, L, E in a row
We can arrange n unique objects in n! ways.
So, we can arrange the 3 letters in 3! ways (= 6 ways)
So, we can complete stage 1 in 6 ways

IMPORTANT: For each arrangement of 3 letters (above), there are 4 places where the two P's can be placed.
For example, in the arrangement AEL, we can add spaces as follows _A_E_L_
So, if we place each P in one of the available spaces, we can ENSURE that the two P's are never together.

Stage 2: Select two available spaces and place an P in each space.
Since the order in which we select the two spaces does not matter, we can use combinations.
We can select 2 spaces from 4 spaces in 4C2 ways (= 6 ways)
So we can complete stage 2 in 6 ways.

By the Fundamental Counting Principle (FCP), we can complete the 2 stages (and thus arrange all 5 letters) in (6)(6) ways (= 36 ways)

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: http://www.gmatprepnow.com/module/gmat-counting/video/775

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

Then you can try solving the following questions:

EASY
- https://www.beatthegmat.com/counting-problem-company-recruitment-t244302.html
- https://www.beatthegmat.com/picking-a-5-digit-code-with-an-odd-middle-digit-t273110.html
- https://www.beatthegmat.com/permutation-combination-simple-one-t257412.html
- https://www.beatthegmat.com/simple-one-t270061.html

MEDIUM
- https://www.beatthegmat.com/combinatorics-solution-explanation-t273194.html
- https://www.beatthegmat.com/arabian-horses-good-one-t150703.html
- https://www.beatthegmat.com/sub-sets-probability-t273337.html
- https://www.beatthegmat.com/combinatorics-problem-t273180.html
- https://www.beatthegmat.com/digits-numbers-t270127.html
- https://www.beatthegmat.com/doubt-on-separator-method-t271047.html
- https://www.beatthegmat.com/combinatorics-problem-t267079.html

DIFFICULT
- https://www.beatthegmat.com/wonderful-p-c-ques-t271001.html
- https://www.beatthegmat.com/permutation-and-combination-t273915.html
- https://www.beatthegmat.com/permutation-t122873.html
- https://www.beatthegmat.com/combinations-t123249.html

Cheers,
Brent

_________________
Brent Hanneson – Creator of GMATPrepNow.com
Use my video course along with

And check out all of these free resources

GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months!

### GMAT/MBA Expert

Legendary Member
Joined
24 Jul 2015
Posted:
2326 messages
Followed by:
32 members
19
GMAT Score:
=>

When we encounter “at least” in counting questions or probability questions, we should consider complementary counting.
The total number of arrangements of the 5 letters is 5!/2! (5! Counts each arrangement of the two Ps 2! times).
The number of arrangements with no letter between the two Ps is 4!.
Thus, the number of arrangements in which at least one letter lies between the two Ps is 5!/2! - 4! = 60 - 24 = 36.

_________________

Math Revolution
Finish GMAT Quant Section with 10 minutes to spare.
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
Only $149 for 3 month Online Course Free Resources-30 day online access & Diagnostic Test Unlimited Access to over 120 free video lessons-try it yourself Email to : info@mathrevolution.com • Free Practice Test & Review How would you score if you took the GMAT Available with Beat the GMAT members only code • Get 300+ Practice Questions 25 Video lessons and 6 Webinars for FREE Available with Beat the GMAT members only code • Magoosh Study with Magoosh GMAT prep Available with Beat the GMAT members only code • Free Trial & Practice Exam BEAT THE GMAT EXCLUSIVE Available with Beat the GMAT members only code • 5-Day Free Trial 5-day free, full-access trial TTP Quant Available with Beat the GMAT members only code • Award-winning private GMAT tutoring Register now and save up to$200

Available with Beat the GMAT members only code

• 1 Hour Free
BEAT THE GMAT EXCLUSIVE

Available with Beat the GMAT members only code

• Free Veritas GMAT Class
Experience Lesson 1 Live Free

Available with Beat the GMAT members only code

• FREE GMAT Exam
Know how you'd score today for \$0

Available with Beat the GMAT members only code

• 5 Day FREE Trial
Study Smarter, Not Harder

Available with Beat the GMAT members only code

### Top First Responders*

1 Ian Stewart 66 first replies
2 Brent@GMATPrepNow 52 first replies
3 Jay@ManhattanReview 52 first replies
4 GMATGuruNY 31 first replies
5 Scott@TargetTestPrep 19 first replies
* Only counts replies to topics started in last 30 days
See More Top Beat The GMAT Members

### Most Active Experts

1 Scott@TargetTestPrep

Target Test Prep

200 posts
2 Max@Math Revolution

Math Revolution

95 posts
3 Brent@GMATPrepNow

GMAT Prep Now Teacher

91 posts
4 Ian Stewart

GMATiX Teacher

72 posts
5 Rich.C@EMPOWERgma...

EMPOWERgmat

67 posts
See More Top Beat The GMAT Experts