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

Question

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

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chetan2u · Accepted Answer

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

EgmatQuantExpert · Answer

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: https://e-gmat.com/blogs/wp-content/uploads/2017/01/628_1.png 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 https://e-gmat.com/blogs/wp-content/uploads/2018/08/Ability-Quiz-AT-e1533292152363.png

EgmatQuantExpert · Answer

The official solution has been posted. Looking forward to a healthy discussion..

ankujgupta · Answer

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;

Abhishek009 · Answer

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

ankujgupta · Answer

chetan2u  Thanks. Got the error.

EgmatQuantExpert · Answer

Method 2 –

4 -letter words in which G,L must be present

two spaces must be filled with the remaining 5 letters

we will clearly see that GL can be arranged in 4P2 ways

the number of ways to arrange GL on 4 spaces = 4P2 = 4!/2! = 12 ways

o E, N, I, S, H

the remaining two spaces can be filled and arranged in = 5P2 = 5!/3! = 20 ways

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

Please Note

it is NOT necessary to place GL together in the 4 letter word

one must avoid falling into such a trap

Thanks,
Saquib
Quant Expert
e-GMAT

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BrentGMATPrepNow · Answer

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.

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GMATGuruNY · Answer

Number of options for G = 4. (Any of the 4 positions in the word.)
Number of options for L = 3. (Any of the 3 remaining positions in the word.)
Number of options for the next position in the word = 5. (Any of the 5 remaining letters.)
Number of options for the last position in the word = 4. (Any of the 4 remaining letters.)
To combine these options, we multiply:
4*3*5*4 = 240

E

satya2029 · Answer

GL and ENISH
there is only one way to select GL together, and for remaining two letters, we can select them in 5c2 ways
=1*5c2=10
and these 4 letters can be arranged in !4 ways
Total=10*!4
=240
E:)

ScottTargetTestPrep · Answer

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

MBAHOUSE · Answer

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

Kinshook · Answer

Asked: 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?

Since G & L are already selected, we have to select 2 letters out of remaining 5 and arrange them.
Number of ways = 5C2 * 4! = 10 * 24 = 240

IMO E

vishalsinghvs08 · Answer

Why can't we do it using the fundamental principle of counting

two places can be filled using a 5x4
the overall arrangement of 4 letters can be accounted for by 4!
therefore 5x4x4!
Not sure what's wrong with this approach

Bunuel ,  KarishmaB

KarishmaB · Answer

You are mixing two approaches. The fundamental counting principle itself gives arrangements. We derive 4! from the fundamental counting principle. What you need to do before that is have 4 distinct letters in hand. You already have G and L so you can select 2 more from the remaining 5 in 5C2 ways. Then you multiply this by 4! Check the fundamental counting principle here: https://youtu.be/LFnLKx06EMU and how we use combinations for selection here: https://youtu.be/tUPJhcUxllQ

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

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