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

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Sat Nov 30, 2019 7:35 pm

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 paired together, with G coming before L and repetition of alphabets being not allowed?

A. 60
B. 120
C. 180
D. 200
E. 240

OA A

Source: e-GMAT

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deloitte247: Legendary Member; Posts: 2214; Joined: Fri Mar 02, 2018 2:22 pmFollowed by: 5 members

by deloitte247 » Sun Dec 01, 2019 5:40 am

The word "ENGLISH" has 7 letters.
We want to select 4-letter words out of which 'G' and 'L' are already chosen. So, we have to choose 2 letters out of the remaining 5 letters.
$$Therefore,\ 5C_2=\frac{5!}{2!\left(5-2\right)!}=\frac{5\cdot4\cdot3!}{2!\cdot3!}=\frac{5\cdot4}{2\cdot1}=10$$
Therefore, 4 letters can be selected in 10 different ways
The 4 letters selected can be arranged in 4! = 4 * 3 * 2 * 1 = 24
Total number of ways in which the 4 letter words can be formed from 'ENGLISH' = 10 * 24 = 240
Answer = option E

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ktrout2020: Senior | Next Rank: 100 Posts; Posts: 41; Joined: Wed Oct 30, 2019 1:10 pm

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

by ktrout2020 » Sun Dec 01, 2019 5:52 am

Are you sure the answer is A?

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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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

by Brent@GMATPrepNow » Sun Dec 01, 2019 6:02 am

BTGmoderatorDC 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 paired together, with G coming before L and repetition of alphabets being not allowed?

A. 60
B. 120
C. 180
D. 200
E. 240

OA A

Source: e-GMAT

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

"Glue" the G and L together, to form the super-letter "GL"
This super-letter (GL) will be combined with the 2 selected letters from stage 1.

Stage 2: Arrange the two letters chosen in stage 1, and the super-letter (GL)
We now basically have 3 characters that we need to arrange.
We can arrange n objects in n! ways
So, we can arrange the 3 characters in 3! ways (= 6 ways)
We can complete stage 2 in 6 ways

By the Fundamental Counting Principle (FCP), we can complete the two stages (and thus create 4-letter words) in (10)(6) ways (= 60 ways)

Answer: A

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Last edited by Brent@GMATPrepNow on Thu Dec 12, 2019 12:56 pm, edited 1 time in total.

Brent Hanneson - Creator of GMATPrepNow.com

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8154; Joined: Sat Apr 25, 2015 10:56 amFollowed by: 29 members; Location: Los Angeles, CA; Thanked: 43 times

by Scott@TargetTestPrep » Sun Dec 08, 2019 7:12 pm

BTGmoderatorDC 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 paired together, with G coming before L and repetition of alphabets being not allowed?

A. 60
B. 120
C. 180
D. 200
E. 240
Source: e-GMAT

Since G and L must be used, the number of ways of choosing 2 more letters from the remaining 5 is 5C2 = (5 x 4)/2 = 10. However, once we have 4 letters, there are 4! = 24 ways to arrange them. Therefore, there are a total of 10 x 24 = 240 words that can be formed.

Answer: E

Scott Woodbury-Stewart
Founder and CEO
[email protected]

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[email protected]: Elite Legendary Member; Posts: 10392; Joined: Sun Jun 23, 2013 6:38 pmFollowed by: 511 members; Location: Palo Alto, CA; Thanked: 2867 times; GMAT Score:800

by [email protected] » Thu Dec 12, 2019 12:00 pm

Hi All,

To start, this question is really poorly-worded, but the math behind it is just "Permutation math." There are a number of different ways to approach it - and even if you don't know how to do all of the math "all at once", you can still solve it by breaking the calculation down into pieces.

We're asked for the number of 4-letter words that can be formed using the letters in the word ENGLISH, with the following restrictions:
1) The letters G and L must be in the word,
2) The G must appear before the L (this is meant to mean that the G appears IMMEDIATELY BEFORE the L, although the way the prompt is written is vague on this point).
3) Letter repetition is NOT allowed.

Since we MUST use the G and the L, we're going to choose 2 additional letters from the 5 remaining letters (E, N, I, S and H). Since the G must come 'immediately before' the L, we can 'map out' the various ways that that can happen:

1st option: G L _ _

Here, the first blank would be filled by one of the 5 other letters (and once we choose a letter, the second blank would be filled with one of the 4 remaining letters). Thus, there are (5)(4) = 20 words that begin with "GL"

From here, we just have to account for all of the ways that the L can immediately follow the G:
G L _ _
_ G L _
_ _ G L

Each of these 3 possibilities would have 20 options each... (3)(20) = 60 possible words

Final Answer: A

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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