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

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Posts: 2303
How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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Updated on: 07 Aug 2018, 00:06
00:00

Difficulty:

65% (hard)

Question Stats:

55% (01:27) correct 45% (01:24) wrong based on 123 sessions

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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
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Originally posted by EgmatQuantExpert on 29 Jan 2017, 23:33.
Last edited by EgmatQuantExpert on 07 Aug 2018, 00:06, edited 1 time in total.
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Joined: 02 Aug 2009
Posts: 7106
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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30 Jan 2017, 08:33
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
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##### General Discussion
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Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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Updated on: 30 Jan 2017, 22:12
The official solution has been posted. Looking forward to a healthy discussion..
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Originally posted by EgmatQuantExpert on 29 Jan 2017, 23:34.
Last edited by EgmatQuantExpert on 30 Jan 2017, 22:12, edited 1 time in total.
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Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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30 Jan 2017, 07:53
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;
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Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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30 Jan 2017, 08:15
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..

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Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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30 Jan 2017, 10:27
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
Joined: 04 Jan 2015
Posts: 2303
How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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Updated on: 07 Aug 2018, 01:34
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:

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
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Originally posted by EgmatQuantExpert on 30 Jan 2017, 22:04.
Last edited by EgmatQuantExpert on 07 Aug 2018, 01:34, edited 1 time in total.
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Joined: 04 Jan 2015
Posts: 2303
Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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30 Jan 2017, 22:12
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

• 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
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Re: How many 4-letter words can be formed using the alphabets of the word  [#permalink]

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11 Dec 2018, 08:48
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)

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

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Re: How many 4-letter words can be formed using the alphabets of the word &nbs [#permalink] 11 Dec 2018, 08:48
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