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.
Hello Guest!
It appears that you are browsing the GMAT Club forum unregistered!
Signing up is free, quick, and confidential.
Join 700,000+ members and get the full benefits of GMAT Club
Registration gives you:
Tests
Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan Prep. All are free for GMAT Club members.
Applicant Stats
View detailed applicant stats such as GPA, GMAT score, work experience, location, application status, and more
Books/Downloads
Download thousands of study notes, question collections, GMAT Club’s Grammar and Math books. All are free!
and many more benefits!
- Register now! It`s easy!
- Already registered? Sign in!
Sep 24
Does GMAT RC seem like an uphill battle? e-GMAT is conducting a masterclass to help you learn reading strategies that can enable you to solve 700+ level RC questions with at least 90% accuracy in less than 10 days.
Sep 25
Attend this webinar to learn a methodical approach to recall rules with ease and score consistently high on GMAT SC questions.
Sep 25
Attend this webinar to learn the core NP concepts and a structured approach to solve 700+ Number Properties questions in less than 2 minutes.
Intern
Joined: 29 Dec 2009
Posts: 35
Location: india
How many words can be formed by taking 4 letters at a time out of the
[#permalink]
00:00
Question Stats:
Hide Show timer Statistics
How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
A. 756
B. 1680
C. 1698
D. 2436
E. 2454
Originally posted by jatt86 on 14 Apr 2010, 04:33.
Last edited by
Bunuel on 26 Feb 2019, 03:23, edited 1 time in total.
Added options.
Math Expert
Joined: 02 Sep 2009
Posts: 86800
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
jatt86 wrote:
1] how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice [double letter].
Selected 4 letters can have following 3 patterns:
1. abcd - all 4 letters are different:
\[8P4=1680\] [choosing 4 distinct letters out of 8, when order matters] or \[8C4*4!=1680\] [choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters];
2. aabb - from 4 letters 2 are the same
and other 2 are also the same:
\[3C2*\frac{4!}{2!2!}=18\] - 3C2 choosing which two double letter will provide two letters [out of 3 double letter - MAT], multiplying by \[\frac{4!}{2!2!}\] to get different arrangements [for example MMAA can be arranged in \[\frac{4!}{2!2!}\] # of ways];
3. aabc - from 4 letters 2 are the same and other 2 are different:
\[3C1*7C2*\frac{4!}{2!}=756\] - 3C1 choosing which letter will proved with 2 letters [out of 3 double letter - MAT], 7C2 choosing
third and fourth letters out of 7 distinct letters left and multiplying by \[\frac{4!}{2!}\] to get different arrangements [for example MMIC can be arranged in \[\frac{4!}{2!}\] # of ways].
1680+18+756=2454
Answer: 2454.
_________________
Intern
Joined: 07 Apr 2010
Posts: 19
Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]
a very similar question:
Find the no: of 4 letter words that can be formed from the string "AABBBBCC" ?
Here we have 3 distinct letters[A,B,C] & 4 slots to fill. What logic do you use to solve this problem?
Math Expert
Joined: 02 Sep 2009
Posts: 86800
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
idiot wrote:
a very similar question:
Find the no: of 4 letter words that can be formed from the string "AABBBBCC"
?
Here we have 3 distinct letters[A,B,C] & 4 slots to fill. What logic do you use to solve this problem?
Three patterns:
1. XXXX - only BBBB, so 1
2. XXYY - 3C2[choosing which will take the places of X and Y from A, B and C]*4!/2!2![arranging]=18
3. XXYZ - 3C1[choosing which will take the place of X from A, B and C]*4!/2![arranging]=36
4. XXXY - 2C1[choosing which will take the place of Y from A and C, as X can be only B]*4!/3![arranging]=8
1+18+36+8=63
_________________
Intern
Joined: 07 Apr 2010
Posts: 19
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
thanks a ton, bunuel
Manager
Joined: 21 Mar 2010
Posts: 94
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
I'm usually not bad with anagram problems like this but the term "words" threw me off completely.
For some reason I assumed
the combination of letters had to combine to make sense, i.e. a "word".
MTHE - is hardly a word, so i started counting actual "words"... so obviously completely bombed the question!
Manager
Joined: 14 Nov 2011
Posts: 105
Location: United States
Concentration: General Management, Entrepreneurship
GPA: 3.61
WE:Consulting [Manufacturing]
Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]
Bunuel wrote:
jatt86 wrote:
1] how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice [double letter].
Selected 4 letters can have following 3 patterns:
1. abcd - all 4 letters are different:
\[8P4=1680\] [choosing 4 distinct letters out of 8, when order matters] or \[8C4*4!=1680\] [choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters];
2. aabb -
from 4 letters 2 are the same and other 2 are also the same:
\[3C2*\frac{4!}{2!2!}=18\] - 3C2 choosing which two double letter will provide two letters [out of 3 double letter - MAT], multiplying by \[\frac{4!}{2!2!}\] to get different arrangements [for example MMAA can be arranged in \[\frac{4!}{2!2!}\] # of ways];
3. aabc - from 4 letters 2 are the same and other 2 are different:
\[3C1*7C2*\frac{4!}{2!}=756\] - 3C1 choosing which letter will proved with 2 letters [out of 3 double
letter - MAT], 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by \[\frac{4!}{2!}\] to get different arrangements [for example MMIC can be arranged in \[\frac{4!}{2!}\] # of ways].
1680+18+756=2454
Answer: 2454.
Hi Bunnel,
Is this a GMAT worthy question?
Math Expert
Joined: 02 Sep 2009
Posts: 86800
Re: How many words can be formed
by taking 4 letters at a time out of the [#permalink]
cumulonimbus wrote:
Bunuel wrote:
jatt86 wrote:
1] how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice [double letter].
Selected 4 letters can have following 3 patterns:
1. abcd - all 4 letters are different:
\[8P4=1680\] [choosing 4 distinct letters out of 8, when order matters] or \[8C4*4!=1680\] [choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct
letters];
2. aabb - from 4 letters 2 are the same and other 2 are also the same:
\[3C2*\frac{4!}{2!2!}=18\] - 3C2 choosing which two double letter will provide two letters [out of 3 double letter - MAT], multiplying by \[\frac{4!}{2!2!}\] to get different arrangements [for example MMAA can be arranged in \[\frac{4!}{2!2!}\] # of ways];
3. aabc - from 4 letters 2 are the same and other 2 are different:
\[3C1*7C2*\frac{4!}{2!}=756\] - 3C1 choosing which letter will proved with 2
letters [out of 3 double letter - MAT], 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by \[\frac{4!}{2!}\] to get different arrangements [for example MMIC can be arranged in \[\frac{4!}{2!}\] # of ways].
1680+18+756=2454
Answer: 2454.
Hi Bunnel,
Is this a GMAT worthy question?
No, but this question is good to practice.
_________________
Intern
Joined: 22 Mar 2013
Posts: 8
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
Bunuel wrote:
jatt86 wrote:
1] how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice [double letter].
Selected 4 letters can have following 3 patterns:
1. abcd - all 4 letters are different:
\[8P4=1680\] [choosing 4 distinct letters out of 8, when order matters] or \[8C4*4!=1680\] [choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters];
2. aabb -
from 4 letters 2 are the same and other 2 are also the same:
\[3C2*\frac{4!}{2!2!}=18\] - 3C2 choosing which two double letter will provide two letters [out of 3 double letter - MAT], multiplying by \[\frac{4!}{2!2!}\] to get different arrangements [for example MMAA can be arranged in \[\frac{4!}{2!2!}\] # of ways];
3. aabc - from 4 letters 2 are the same and other 2 are different:
\[3C1*7C2*\frac{4!}{2!}=756\] - 3C1 choosing which letter will proved with 2 letters [out of 3 double
letter - MAT], 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by \[\frac{4!}{2!}\] to get different arrangements [for example MMIC can be arranged in \[\frac{4!}{2!}\] # of ways].
1680+18+756=2454
Answer: 2454.
Bunuel, this is a damn hard question and I find myself not fully able to understand your logic. I am from a very weak background but I have poured through all of the MGMAT math books [excluding the advanced one] several times and still find myself unable to intutively figure out the steps to this problem.
What extra review would you suggest so I can be able to at least follow your solutions to these answers?
Math Expert
Joined: 02 Sep 2009
Posts: 86800
Re:
How many words can be formed by taking 4 letters at a time out of the [#permalink]
tmipanthers wrote:
Bunuel wrote:
jatt86 wrote:
1] how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice [double letter].
Selected 4 letters can have following 3 patterns:
1. abcd - all 4 letters are different:
\[8P4=1680\] [choosing 4 distinct letters out of 8, when order matters] or \[8C4*4!=1680\] [choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct
letters];
2. aabb - from 4 letters 2 are the same and other 2 are also the same:
\[3C2*\frac{4!}{2!2!}=18\] - 3C2 choosing which two double letter will provide two letters [out of 3 double letter - MAT], multiplying by \[\frac{4!}{2!2!}\] to get different arrangements [for example MMAA can be arranged in \[\frac{4!}{2!2!}\] # of ways];
3. aabc - from 4 letters 2 are the same and other 2 are different:
\[3C1*7C2*\frac{4!}{2!}=756\] - 3C1 choosing which letter will proved with 2
letters [out of 3 double letter - MAT], 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by \[\frac{4!}{2!}\] to get different arrangements [for example MMIC can be arranged in \[\frac{4!}{2!}\] # of ways].
1680+18+756=2454
Answer: 2454.
Bunuel, this is a damn hard question and I find myself not fully able to understand your logic. I am from a very weak background but I have poured through all of the MGMAT math books [excluding the advanced one] several times and still find myself unable to intutively figure out the steps to this problem.
What extra review would you suggest so I can be able to at least follow your solutions to these answers?
This question is out of the scope of the GMAT, so I wouldn't worry about it too much.
As for the recommendations.
Best GMAT Books: best-gmat-math-prep-books-reviews-recommendations-77291.html
Theory on Combinations: math-combinatorics-87345.html
DS questions on Combinations: search.php?search_id=tag&tag_id=31
PS
questions on Combinations: search.php?search_id=tag&tag_id=52
Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html
Hope it helps.
_________________
Intern
Joined: 27 Mar 2013
Posts: 42
Location: United States
Concentration: Strategy, Entrepreneurship
GPA: 3.25
WE:General Management [Energy and Utilities]
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
If this would have been a word with three of the same letter I'm assuming you would have more than 3 combinations?
Thanks!
Math Expert
Joined: 02 Sep 2009
Posts: 86800
Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]
Mbearmann wrote:
If this would have been a word with three of the same letter I'm assuming you would have more than 3 combinations?
Thanks!
Yes, we would have one more combination {a, a, a, b}.
_________________
Intern
Joined: 04 Apr 2015
Posts: 13
Concentration: Human Resources, Healthcare
GMAT Date: 08-06-2015
GPA: 3.83
WE:Editorial and Writing [Journalism and Publishing]
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
I can't understand this question. Why is this a combination question and not permutation? isnt it asking for arrangements?
SVP
Joined: 20 Mar 2014
Posts: 2426
Concentration: Finance, Strategy
GMAT 1: 750 Q49 V44
GPA: 3.7
WE:Engineering [Aerospace and Defense]
Re:
How many words can be formed by taking 4 letters at a time out of the [#permalink]
ddg wrote:
I can't understand this question. Why is this a combination question and not permutation? isnt it asking for arrangements?
You are half right.
Permutations = Combinations * n! [where n is the number of 'elements']. In this question, you first need to select the letters out of the given one [combination implied as selection = combination!!] and only after you have selected the letters , you can look at the arrangements. You can not directly go to arrangements as you need to follow the 2 step process:
1. Choose 4 out of 11 letters
2. Arrangement of those selections of 4 letters to
get all the possible arrangements.
Your approach would have been correct, had the question ask us to arrange all of these 11 letters into words of 11 letters or if all the letters were different.
_________________
Intern
Joined: 04 Apr 2015
Posts: 13
Concentration: Human Resources, Healthcare
GMAT Date: 08-06-2015
GPA: 3.83
WE:Editorial and Writing [Journalism and Publishing]
Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]
Thanks!
GMAT Club Legend
Joined: 03 Jun 2019
Posts: 4773
Location: India
GMAT 1: 690 Q50 V34
WE:Engineering [Transportation]
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
jatt86 wrote:
How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
A. 756
B. 1680
C. 1698
D. 2436
E. 2454
Asked: How many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
M-2
A-2
T-2
H-1
E-1
I-1
C-1
S-1
Words of the form abcd = \[^8C_4 * 4! = 1680\]
Words of the form aabc = \[^3C_1*^7C_2* 4!/2! = 3*21*12 = 756\]
Words of the form aabb = \[^3C_2 * 4!/2!/2! = 3* 24/4 = 18\]
Number of words that can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS = 1680+756+18=2454
IMO E
_________________
Kinshook Chaturvedi
Email:
Senior Manager
Joined: 30 Jun 2019
Posts: 286
Re: How many words
can be formed by taking 4 letters at a time out of the [#permalink]
I understand the solution, but where does this logic break down?
MATHEMATICS = 11 letters
MM AA TT = 3 groups of 2
repeats
[11*10*9*8]/[2!2!2!] = 990
Intern
Joined: 25 Jan 2020
Posts: 1
Re: How many words can be formed by taking 4 letters at a time out of the
[#permalink]
Bunuel wrote:
jatt86 wrote:
1] how many words can be formed by taking 4 letters at a time out of the letters of the word MATHEMATICS.
There are 8 distinct letters: M-A-T-H-E-I-C-S. 3 letters M, A, and T are represented twice [double letter].
Selected 4 letters can have following 3 patterns:
1. abcd - all 4 letters are different:
\[8P4=1680\] [choosing 4 distinct letters out of 8, when order matters] or \[8C4*4!=1680\] [choosing 4 distinct letters out of 8 when order does not matter and multiplying by 4! to get different arrangement of these 4 distinct letters];
2. aabb -
from 4 letters 2 are the same and other 2 are also the same:
\[3C2*\frac{4!}{2!2!}=18\] - 3C2 choosing which two double letter will provide two letters [out of 3 double letter - MAT], multiplying by \[\frac{4!}{2!2!}\] to get different arrangements [for example MMAA can be arranged in \[\frac{4!}{2!2!}\] # of ways];
3. aabc - from 4 letters 2 are the same and other 2 are different:
\[3C1*7C2*\frac{4!}{2!}=756\] - 3C1 choosing which letter will proved with 2 letters [out of 3 double
letter - MAT], 7C2 choosing third and fourth letters out of 7 distinct letters left and multiplying by \[\frac{4!}{2!}\] to get different arrangements [for example MMIC can be arranged in \[\frac{4!}{2!}\] # of ways].
1680+18+756=2454
Answer: 2454.
hey! Why are we taking 4C2 in the aabb combination? If we are looking to calculate how the letters are arranged, shouldn't we be using 4P2 instead?
Thanks
Re: How many words can be formed by taking 4 letters at a time out of the [#permalink]
27 Feb 2020, 17:50
Moderators:
Senior Moderator - Masters Forum
3086 posts