Last visit was: 24 Jun 2025, 14:55 It is currently 24 Jun 2025, 14:55
Close
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Close
Request Expert Reply
Confirm Cancel
User avatar
goodyear2013
Joined: 21 Oct 2013
Last visit: 29 May 2020
Posts: 390
Own Kudos:
5,466
 [19]
Given Kudos: 289
Posts: 390
Kudos: 5,466
 [19]
3
Kudos
Add Kudos
16
Bookmarks
Bookmark this Post
Most Helpful Reply
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 24 Jun 2025
Posts: 102,292
Own Kudos:
735,211
 [6]
Given Kudos: 94,011
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,292
Kudos: 735,211
 [6]
4
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
General Discussion
avatar
MelanieMa
Joined: 28 Feb 2014
Last visit: 16 Sep 2014
Posts: 5
Own Kudos:
Given Kudos: 3
Posts: 5
Kudos: 3
Kudos
Add Kudos
Bookmarks
Bookmark this Post
User avatar
MacFauz
Joined: 02 Jul 2012
Last visit: 19 Mar 2022
Posts: 997
Own Kudos:
Given Kudos: 116
Location: India
Concentration: Strategy
GMAT 1: 740 Q49 V42
GPA: 3.8
WE:Engineering (Energy)
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Fairly straightforward question. No reason for it to not be "GMAT style"

a) Three letters give : 3 + 3C2 = 3 + 3 = 6

b) Four letters give : 4 + 4C2 = 4 + 12 = 16

So, answer is B
User avatar
ind23
Joined: 10 Apr 2014
Last visit: 02 Jul 2015
Posts: 22
Own Kudos:
Given Kudos: 3
Posts: 22
Kudos: 49
Kudos
Add Kudos
Bookmarks
Bookmark this Post
MacFauz
Fairly straightforward question. No reason for it to not be "GMAT style"

a) Three letters give : 3 + 3P2 = 3 + 6 = 9
So, we will need more than 3
So, answer is B

Hello - Just a quick thing, the question says "the order of the letter does not matter", so we should be using combinations and permutations.
User avatar
MacFauz
Joined: 02 Jul 2012
Last visit: 19 Mar 2022
Posts: 997
Own Kudos:
3,304
 [1]
Given Kudos: 116
Location: India
Concentration: Strategy
GMAT 1: 740 Q49 V42
GPA: 3.8
WE:Engineering (Energy)
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Oops.. Sorry.. My mistake.. Edited above post.. :)

Thanks

Posted from my mobile device
User avatar
LakerFan24
Joined: 26 Dec 2015
Last visit: 03 Apr 2018
Posts: 167
Own Kudos:
673
 [1]
Given Kudos: 1
Location: United States (CA)
Concentration: Finance, Strategy
WE:Investment Banking (Finance: Venture Capital)
Posts: 167
Kudos: 673
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
This looks like a pattern question. i tried to just understand what they were asking and picked out some letters to test:

if 1 letter, only allowed 1 code (i.e. A)

if 2 letters, allowed 3 codes (i.e. A, B, AB)

if 3 letters, allowed 6 codes (i.e. A, B, C, AB, AC, BC)

if 4 letters, allowed 10 codes (i.e. A, B, C, D, AB, AC, AD, BC, BD, CD)
User avatar
attari92
Joined: 25 Apr 2016
Last visit: 28 May 2019
Posts: 56
Own Kudos:
Given Kudos: 308
Posts: 56
Kudos: 42
Kudos
Add Kudos
Bookmarks
Bookmark this Post
three letters could be enough for uniquely naming 9 books i.e (3! *3) and remaining 1 could be named by taking one more letter under consideration.
Least total letters would be 10
User avatar
Cinematiccuisine
Joined: 02 Sep 2018
Last visit: 05 Jul 2020
Posts: 55
Own Kudos:
Given Kudos: 64
Location: United States
WE:Information Technology (Computer Software)
Posts: 55
Kudos: 195
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Questions:
1) How do i rephrase this question? The Kaplan answer is not so easy to understand ( at least for me)
2) without getting into too many calculations, how can i solve this?

Thank you for your help.
User avatar
MagooshExpert
User avatar
Magoosh GMAT Instructor
Joined: 30 Oct 2017
Last visit: 15 Jan 2020
Posts: 231
Own Kudos:
429
 [2]
Given Kudos: 20
Expert
Expert reply
Posts: 231
Kudos: 429
 [2]
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Cinematiccuisine
Amy is organizing her bookshelves and finds that she has 10 different types of books. She then codes each book with either a single letter or a pair of two different letters. If each type of book is uniquely represented by either a single letter or pair of letters, what is the smallest number of letters Amy will need to create the codes for all 10 types of books? (Assume the order of letters in a pair does not matter.)

A) 3
B) 4
C) 5
F) 10
E) 20

Questions:
1) How do i rephrase this question? The Kaplan answer is not so easy to understand ( at least for me)
2) without getting into too many calculations, how can i solve this?

Thank you for your help.
Hi Cinematiccuisine!

Happy to help :-)

Since the numbers we're dealing with here are small, it is probably easiest to just solve this manually, at first, at least to understand what the question is asking about. Amy is labeling each type of book with either one or two letters. If she is using just one letter, this means there is only one code she can use:

A

If she is using two letters, then there are three:

A
B
AB

We are told that the order of the letters doesn't matter, so we don't have to worry about counting AB and BA separately. Now, with three letters, we have all of the above combinations, plus three more:

C
AC
BC

That's six total. If we add one more, then we have those six plus four more:

D
AD
BD
CD

And that give us 10 possible codes, which is what the question is asking for :-) So the answer is 4.

Now, if the number was much bigger, then we might need to create a formula to solve this. The easiest way to do that is to just notice the pattern. Every time we add a new letter, we get one more code (just that letter), plus the number of letters that we already have, which create the combinations of mixed letters (like AD, BD, CD). So adding C gave us 3 more codes, adding D gives us 4 more codes, adding E gives us 5 more codes, and so on. We can use that pattern to determine the number of possible codes at pretty much any step :-)

We can write this out in terms of combinations, too. If N is the number of letters that we're using, then the number of codes can be written as:

N + NC2

So with four letters, we get 4 + 4C2 = 10 different codes.

I hope that helps! :-)
-Carolyn
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 22 Jun 2025
Posts: 11,304
Own Kudos:
41,369
 [1]
Given Kudos: 333
Status:Math and DI Expert
Products:
Expert
Expert reply
Posts: 11,304
Kudos: 41,369
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Cinematiccuisine
Amy is organizing her bookshelves and finds that she has 10 different types of books. She then codes each book with either a single letter or a pair of two different letters. If each type of book is uniquely represented by either a single letter or pair of letters, what is the smallest number of letters Amy will need to create the codes for all 10 types of books? (Assume the order of letters in a pair does not matter.)

A) 3
B) 4
C) 5
F) 10
E) 20

Questions:
1) How do i rephrase this question? The Kaplan answer is not so easy to understand ( at least for me)
2) without getting into too many calculations, how can i solve this?

Thank you for your help.

Please put the first few words as the topic name.

What it means?
10 books have to be given a code as a single letter such as A, B etc or as a double letter such as AB, BA, AC etc. AB and BA will be different as both are different as code

Solution
Let the number of letters required is n,
So single digits will be n
Two digits will be nC2........
.we are not multiplying with 2! because order does not matter.
So \(n+nC2\geq{10}\)........\(n+\frac{n!}{(n-2)!2!}\geq{10}........n+n(n-1)/2\geq{10}...............(2n+n^2-n}\geq{20}............n(n+1)\geq{20}.........n(n+1)\geq{4*5}..\)
So minimum value = 4

B
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 24 Jun 2025
Posts: 102,292
Own Kudos:
Given Kudos: 94,011
Products:
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,292
Kudos: 735,211
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Cinematiccuisine
Amy is organizing her bookshelves and finds that she has 10 different types of books. She then codes each book with either a single letter or a pair of two different letters. If each type of book is uniquely represented by either a single letter or pair of letters, what is the smallest number of letters Amy will need to create the codes for all 10 types of books? (Assume the order of letters in a pair does not matter.)

A) 3
B) 4
C) 5
F) 10
E) 20

Questions:
1) How do i rephrase this question? The Kaplan answer is not so easy to understand ( at least for me)
2) without getting into too many calculations, how can i solve this?

Thank you for your help.

Merging topics.

Please follow the rules when posting a question: https://gmatclub.com/forum/rules-for-po ... 33935.html
User avatar
VodkaHelps
Joined: 01 Nov 2017
Last visit: 05 Apr 2020
Posts: 76
Own Kudos:
Given Kudos: 171
GMAT 1: 640 Q49 V28
GMAT 2: 700 Q50 V35
GMAT 3: 680 Q47 V36
GPA: 3.84
Products:
GMAT 3: 680 Q47 V36
Posts: 76
Kudos: 82
Kudos
Add Kudos
Bookmarks
Bookmark this Post
The question asks what is the smallest number of letters required to create at least 10 unique codes. Because the question told us that the books are numbered by EITHER a single letter or 2 different letters, and we need to find the least number of letters to create 10 codes, we can safely look into the situation in which all books are numbered by 2 different letters.

Let's call "n" is the number of letters we need to find, then we have

n x n(-1) >= 10
4 x 3 =12 -> MORE than 10 -> keep this
3 x 2 = 6 -> LESS than 10 -> discard
5 x 4 = 20 -> MORE than 10 -> but n=5 < 4 -> we pick the n=4
User avatar
HardikL
Joined: 09 Jun 2016
Last visit: 04 Feb 2024
Posts: 11
Own Kudos:
Given Kudos: 4
GMAT 1: 710 Q48 V39
GMAT 2: 730 Q49 V39
GMAT 2: 730 Q49 V39
Posts: 11
Kudos: 11
Kudos
Add Kudos
Bookmarks
Bookmark this Post
For these type of questions, let us assume that n distinct letters/digits are required.

then for second code of two digits numbers can be selected as nC2

thus we get total number of options :
10 = n+ nC2
=> 2n + n(n-1) = 20
=> n^2-n-20=0
=> (n-5)(n+4)=0
or n=5 as n cannot be negative.
User avatar
Archit3110
User avatar
Major Poster
Joined: 18 Aug 2017
Last visit: 24 Jun 2025
Posts: 8,243
Own Kudos:
Given Kudos: 243
Status:You learn more from failure than from success.
Location: India
Concentration: Sustainability, Marketing
GMAT Focus 1: 545 Q79 V79 DI73
GPA: 4
WE:Marketing (Energy)
GMAT Focus 1: 545 Q79 V79 DI73
Posts: 8,243
Kudos: 4,752
Kudos
Add Kudos
Bookmarks
Bookmark this Post
total letters required ;
a,b,c,d; 4
pair can be made
ab,ac,ad,bc,bd,cd,a,b,c,d
IMO B


goodyear2013
Amy is organizing her bookshelves and finds that she has 10 different types of books. She then codes each book with either a single letter or a pair of two different letters. If each type of book is uniquely represented by either a single letter or pair of letters, what is the smallest number of letters Amy will need to create the codes for all 10 types of books? (Assume the order of letters in a pair does not matter.)

A) 3
B) 4
C) 5
D) 10
E) 20

OE:
The question asks for the smallest value of n, such that (n + nC2) = 10 (n represents the number of letters. In this equation, n by itself is for single-letter codes and nC2 is for two-letter codes).

At this point, you'd need to pick numbers, since there's really no easy way to solve nC2 = (10 – n) without a calculator.
Looking at the answer choices, you can eliminate 10 and 20, so you can quickly narrow down the values you need to test. (i.e. (10 – n) suggests n can not be greater than 10.)

As a general rule, whenever you're asked for the smallest value that satisfies a condition, start by testing the smallest number in the answers. Conversely, if you're asked for the largest value, start with the greatest answer.
Plug-in n=4 to (n + nC2) = (4 + 4C2) = 4 + (4x3 /2) = (4 + 6) = 10

Hi, I want to know whether this reverse combination calculation is the GMAT style questions, please.
I have not seen one in the Official question.
User avatar
CEdward
Joined: 11 Aug 2020
Last visit: 14 Apr 2022
Posts: 1,212
Own Kudos:
Given Kudos: 332
Posts: 1,212
Kudos: 244
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Answer is B.

A
B
AB
C
AC
BC
BA
CA
CB
D
User avatar
ScottTargetTestPrep
User avatar
Target Test Prep Representative
Joined: 14 Oct 2015
Last visit: 24 Jun 2025
Posts: 20,993
Own Kudos:
26,044
 [1]
Given Kudos: 293
Status:Founder & CEO
Affiliations: Target Test Prep
Location: United States (CA)
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 20,993
Kudos: 26,044
 [1]
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
goodyear2013
Amy is organizing her bookshelves and finds that she has 10 different types of books. She then codes each book with either a single letter or a pair of two different letters. If each type of book is uniquely represented by either a single letter or pair of letters, what is the smallest number of letters Amy will need to create the codes for all 10 types of books? (Assume the order of letters in a pair does not matter.)

A) 3
B) 4
C) 5
D) 10
E) 20

Solution:

Since there are only 10 books, the number of colors needed is even fewer. Let’s check the answer choices starting with 3.

If there are 3 colors, the number of single-color codes is 3 and the number of codes that are pairs of two different colors is 3C2 = (3 x 2)/2 = 3. Therefore, if there are 5 colors, the total number of codes can be formed is 3 + 3 = 6. However, this is not sufficient since there are 10 books.

If there are 4 colors, the number of single-color codes is 4 and the number of codes that are pairs of two different colors is 4C2 = (4 x 3)/2 = 6. Therefore, if there are 4 colors, the total number of codes can be formed is 4 + 6 = 10. We see that this is sufficient since there are exactly 10 books.

Answer: B
User avatar
shisingh
Joined: 25 May 2024
Last visit: 15 Jun 2025
Posts: 24
Own Kudos:
Given Kudos: 39
Location: India
Concentration: Economics, Strategy
Schools: ISB '27
GMAT Focus 1: 665 Q84 V85 DI80
Schools: ISB '27
GMAT Focus 1: 665 Q84 V85 DI80
Posts: 24
Kudos: 1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
I think approach is right but 2n + n(n-1) = n^2+n so solution will be n = 4

HardikL
For these type of questions, let us assume that n distinct letters/digits are required.

then for second code of two digits numbers can be selected as nC2

thus we get total number of options :
10 = n+ nC2
=> 2n + n(n-1) = 20
=> n^2-n-20=0
=> (n-5)(n+4)=0
or n=5 as n cannot be negative.
Moderators:
Math Expert
102292 posts
PS Forum Moderator
657 posts