Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

It is currently 20 Jul 2019, 04:51

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

How many 4-letter words can be formed using the alphabets of the word

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 2943
How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post Updated on: 07 Aug 2018, 01:06
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

55% (01:57) correct 45% (02:02) wrong based on 144 sessions

HideShow timer Statistics


How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?


    A. \(60\)
    B. \(120\)
    C. \(180\)
    D. \(200\)
    E. \(240\)

Thanks,
Saquib
Quant Expert
e-GMAT

To read all our articles: Must read articles to reach Q51


Image

_________________

Originally posted by EgmatQuantExpert on 30 Jan 2017, 00:33.
Last edited by EgmatQuantExpert on 07 Aug 2018, 01:06, edited 1 time in total.
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Aug 2009
Posts: 7764
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post 30 Jan 2017, 09:33
4
1
EgmatQuantExpert wrote:
How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?


    A. \(60\)
    B. \(120\)
    C. \(180\)
    D. \(200\)
    E. \(240\)

Thanks,
Saquib
Quant Expert
e-GMAT


Hi ankujgupta
Do not start arranging the alphabets before selection.
Total 7 letters out of which 4 are to be choosen..
G and L are already there, so choose 2 out of remaining 5.. 5C2=\(\frac{5!}{3!2!}=10\)..

Now 4 letters can be choosen in 10 ways but they can be arranged in 4! Or 4*3*2=24 ways..

Total 10*24=240..
E
_________________
General Discussion
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 2943
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post Updated on: 30 Jan 2017, 23:12

Originally posted by EgmatQuantExpert on 30 Jan 2017, 00:34.
Last edited by EgmatQuantExpert on 30 Jan 2017, 23:12, edited 1 time in total.
Manager
Manager
avatar
S
Joined: 21 Jan 2016
Posts: 77
Location: India
GMAT 1: 670 Q50 V30
WE: Engineering (Computer Software)
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post 30 Jan 2017, 08:53
Is answer C ? We need to select 2 alphabets from E,N,I,S and H, which are arranged also, so 5P2 = 20. We have 3 places so total = 20*3*2*1 = 120;
Board of Directors
User avatar
P
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4512
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post 30 Jan 2017, 09:15
1
EgmatQuantExpert wrote:
How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?


    A. \(60\)
    B. \(120\)
    C. \(180\)
    D. \(200\)
    E. \(240\)



We have here 7 letters ( E,N,G,L,I,S & H )

And 4 places as _ _ _ _

G & L must be there, so we can arrange the 2 letters in 2! or 2 ways and the next 5 Letters ( E,N,I,S & H ) in 2 places in 5! ways ie, 120 ways...

So, Total Number of ways is 120*2 = 240 ways..

Answer must be (E) 240

_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
Manager
Manager
avatar
S
Joined: 21 Jan 2016
Posts: 77
Location: India
GMAT 1: 670 Q50 V30
WE: Engineering (Computer Software)
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post 30 Jan 2017, 11:27
chetan2u wrote:
EgmatQuantExpert wrote:
How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?


    A. \(60\)
    B. \(120\)
    C. \(180\)
    D. \(200\)
    E. \(240\)

Thanks,
Saquib
Quant Expert
e-GMAT


Hi ankujgupta
Do not start arranging the alphabets before selection.
Total 7 letters out of which 4 are to be choosen..
G and L are already there, so choose 2 out of remaining 5.. 5C2=\(\frac{5!}{3!2!}=10\)..

Now 4 letters can be choosen in 10 ways but they can be arranged in 4! Or 4*3*2=24 ways..

Total 10*24=240..
E

chetan2u Thanks. Got the error.
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 2943
How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post Updated on: 07 Aug 2018, 02:34
Hey,

PFB the official solution. :)

This question can be done in a number of ways. Let us focus on the two important methods of solving this question –

Method 1 –

Step 1: Understand the objective

The objective of the question is to find the number of 4-letter words that can be formed from the alphabets of the word ENGLISH.

The information given is:

    • There are a total of 7 alphabets: E, N, G, L, I, S and H.
    • Repetition of letters is not allowed.
    • Two letters G and L are to be included necessarily in all the words.
    • Per the question, it is not necessary for the word formed to convey a meaning per English dictionary.


So, now we know the objective of the question and the information provided in the question.

Step 2: Write the objective equation enlisting all tasks

The objective comprises of the following tasks:

    • Task 1 – Select two letters from the letters G and L. (As these two are to be necessarily included)
    • Task 2 – Select two letters from E, N, I, S, and H.
    • Task 3 – Form 4-letter words from the four letters that are selected in the previous two tasks.
      o Now, in order to accomplish the objective, all the three tasks need to be done. So, in the objective equation, we will put a multiplication sign between the number of ways of doing the three tasks.

The Objective Equation will therefore be:

Image

Step 3: Determine the number of ways of doing each task

    • Task 1 is to select two letters from the letters G and L.

      o Now, the number of ways to select 2 letters from 2 different letters = 2C2 = 1
      o Thus, number of ways to do Task 1 = 1

    • Task 2 is to select two letters from the 5 letters E, N, I, S, and H.

      o The number of ways to select 2 letters from 5 different letters = 5C2 = 10
      o Thus, number of ways to do Task 2 = 10

    • Task 3 is to arrange the selected 4 letters in 4 spaces to form different words.

      o Now, the number of ways in which 4 letters can be arranged in 4 spaces= 4! = 4 X 3 X 2 X 1
      o Thus, number of ways to do Task 3 = 24

Step 4: Calculate the final answer

    • In this step, we are going to plug the values in the above equation:
      o Number of different 3-letter words = 1 x 10 x 24 = 240
      o So, there are 240 words that can be formed per the condition stated in the question.
    • Thus, the correct answer choice is Option D.


Thanks,
Saquib
Quant Expert
e-GMAT


Image

_________________

Originally posted by EgmatQuantExpert on 30 Jan 2017, 23:04.
Last edited by EgmatQuantExpert on 07 Aug 2018, 02:34, edited 1 time in total.
e-GMAT Representative
User avatar
V
Joined: 04 Jan 2015
Posts: 2943
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post 30 Jan 2017, 23:12
1
Method 2 –


    • Now let us look at another way to solve this question.
    • We need to make 4 -letter words in which G,L must be present .
    • And the remaining two spaces must be filled with the remaining 5 letters.
    • Thus, the objective equation can be written as

Image

    • Thus if we make 4 spaces and try to arrange GL in these 4 spaces, we will clearly see that GL can be arranged in 4P2 ways.
      o Thus, the number of ways to arrange GL on 4 spaces = 4P2 = 4!/2! = 12 ways.

    • To fill the remaining two spaces, we have 5 letters to choose from and arrange them
      o E, N, I, S, H

    • Thus, the remaining two spaces can be filled and arranged in = 5P2 = 5!/3! = 20 ways.
    • Plugging the two values in the objective equation, we get

      o The number of ways to make 4 letter words = 12 x 20 = 240 ways.

    • As we can see both the methods give us the same answer. In the first method, we first selected the letters and then arranged them and in the second method, we did the selection and arrangement simultaneously.

    Please Note: When we say that G and L must be there in the 4 letter word, it means that they can be placed anywhere in the 4 letter word. And it is NOT necessary to place GL together in the 4 letter word. This is a common mistake, which a lot of students make and one must avoid falling into such a trap!


Thanks,
Saquib
Quant Expert
e-GMAT

Register for our Free Session on Number Properties (held every 3rd week) to solve exciting 700+ Level Questions in a classroom environment under the real-time guidance of our Experts :)

Image
_________________
CEO
CEO
User avatar
V
Joined: 12 Sep 2015
Posts: 3852
Location: Canada
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

Show Tags

New post 11 Dec 2018, 09:48
Top Contributor
EgmatQuantExpert wrote:
How many 4-letter words can be formed using the alphabets of the word ENGLISH, if it is given that the 4-letter word contains alphabets G and L and repetition of alphabets are not allowed?

    A. \(60\)
    B. \(120\)
    C. \(180\)
    D. \(200\)
    E. \(240\)


Take the task of creating 4-letter words and break it into stages.

Stage 1: Select 2 letters from E, N, I, S, H
Since the order in which we select the two letters does not matter (yet!!), we can use combinations.
We can select 2 letters 5 letters in 5C2 ways (10 ways)
So, we can complete stage 1 in 10 ways

ASIDE: If anyone is interested, we have a video on calculating combinations (like 5C2) in your head below

Stage 2: Combine G and L with the two letters you chose in stage 1, and then arrange those 4 letters
We can arrange n objects in n! ways
So, we can arrange the 4 letters in 4! ways (= 24 ways)
We can complete stage 2 in 24 ways

By the Fundamental Counting Principle (FCP), we can complete the two stages (and thus create 4-letter words) in (10)(24) ways (= 240 ways)

Answer: E

Note: the FCP can be used to solve the MAJORITY of counting questions on the GMAT. So, be sure to learn it.

RELATED VIDEOS





_________________
Test confidently with gmatprepnow.com
Image
GMAT Club Bot
Re: How many 4-letter words can be formed using the alphabets of the word   [#permalink] 11 Dec 2018, 09:48
Display posts from previous: Sort by

How many 4-letter words can be formed using the alphabets of the word

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne