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

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

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;

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

chetan2u Thanks. Got the error.

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
e-GMAT

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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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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

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:)

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

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

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

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

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