In how many ways can a 4-letter word be formed from the letters ABCDEF

Question

In how many ways can a 4-letter word be formed from the letters ABCDEFIO such that the word contains 2 vowels and 2 consonants? The letters cannot be repeated, and, the words can have no dictionary meaning.

A) 36
B) 144
C) 288
D) 864
E) 1728

A) 36
B) 144
C) 288
D) 864
E) 1728

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kunalcvrce · Accepted Answer

AEIO- 2 vowels can be selected in 4C2 ways.
BCDF- 2 consonant can be selected 4C2 ways.
we can arrannge 4 alphabets in 4! ways.

so total ways=4C2 *4C2 *4!=864

option D

EgmatQuantExpert · Answer

Solution

Given:  
 • We are given 8 letters- ABCDEFIO
• We need to form 4 letter words from ABCDEFIO such that it contains 2 vowels and 2 consonants
• No letter in the 4-letter word should be repeated

To find:

• We need to find the number of ways in which 4-letter word can be formed such that it includes 2 vowels and 2 consonants from the 8 letters- ABCDEFIO

Approach and Working:

The 8 letters- ABCDEFIO has 4 consonants and 4 vowels.
Hence, we first we will pick 2 consonants from 4 consonants and 2 vowels from 4 vowels to form the 4-letter word.

Thus, 
 • Total ways= ways to pick 2 consonants from 4 consonants * ways to pick 2 vowels from 4 vowels* ways to fill the four places

Ways to picks 2 consonants from 4 consonants:

• 2 consonants from the 4 consonants can be picked in 4c2=6 ways

Ways to pick 2 vowels from 4 vowels:

• 2 vowels from the 4 vowels can be picked in 4c2=6 ways.

Ways to fill the 4 places in the 4-letter word:

• Total ways to fill the 4 places= Ways to fill the 1st place* ways to fill the 2nd place* ways to fill the 3rd place* ways to fill the 4th place.
 
Since there are only 4 letters, the place can at max be filled with 4 letters.
 • Hence, ways to fill the first place= 4

Now, there will be 3- letters to fill the 2nd place. Hence,
 • Ways to fill the 2nd place= 3

In the similar fashion, 
 • Ways to fill the 3rd place= 2
• Ways to fill the 1st place= 1

Hence, total ways to form the 4 letters word= 6*6*4*3*2*1= 864

Hence, the correct answer is option D.
 
  Answer: D

monish447 · Answer

while solving the question, the statement "the words can have no dictionary meaning" has not been accounted for.

EgmatQuantExpert · Answer

Hey monish447 , If we consider a word like ABCE or DFEO - both these words satisfy the given condition in the question. However, none of them are meaningful words that you find in the dictionary. Hope this answers your query.

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monish447 · Answer

True.
But how about the word "FADE"
As per your solution you have taken this as a part of the solution but It must be excluded as per the ques.

EgmatQuantExpert · Answer

In this case let me point out to the specific language given in the question. The question says "the words can have no dictionary meaning" - 'can have' doesn't necessarily mean it is always true. What you are saying would have been true if the question was given like "the words must have no dictionary meaning" - the word 'must' ensures that every word is not a dictionary word. Hope this clarifies your doubt.

fskilnik · Answer

?\,\,:\,\,\# \,\,4\,\,{\rm{distinct}}\,\,{\rm{letters}}\,\,\,\left\{ \matrix{
 \,2\,\,{\rm{vowels}} \hfill \cr 
 \,2\,\,{\rm{consonants}} \hfill \cr} \right.

?\,\,\, = \,\,\,\underbrace {C\left( {4,2} \right)}_{{\rm{for}}\,\,{\rm{vowels}}} \cdot \underbrace {C\left( {4,2} \right)}_{{\rm{for}}\,\,{\rm{consonants}}} \cdot \underbrace {\,\,{P_4}\,\,}_{{\rm{chosen}}\,4\,,\,\,{\rm{permutations}}}\,\,\, = \,\,\,{{4 \cdot 3} \over 2} \cdot {{4 \cdot 3} \over 2} \cdot 4!\,\,\, = \,\,\,6 \cdot 6 \cdot 24\,\,\, = \,\,\,864

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

Archit3110 · Answer

C= BCDF
V= AEIO
4c2*4c2*4! = 864
IMOD

chetan2u · Answer

‘ the words can have no dictionary meaning.’ should be ‘ the words need not have a dictionary meaning.’

Solution:
There are 4 vowels and 4 consonants.

Step I : Choose 2 from each 
Vowels- 4C2
Consonants- 4C2
 Each combination of vowel can take combination of consonants, so 4C2*4C2 or 6*6 or 36 ways.

Step II : Each of these selected can be arranged in 4! ways. => 36*4! = 36*24

D

saumitrag · Answer

Why cant we use the approach P(4,2)   * P(4,2)  . Cause the order in selection should ideally matter while selecting the numbers,

GGGMAT2 · Answer

The word 'CODE' has dictionary meaning, and is included in the count. How do we exclude such words from the count?

chetan2u · Answer

Hi

The GMAT will never test you about possible English dictionary words in quant.

Such questions would have words 'CAN have no dictionary meaning.', which would mean all possible words irrespective of those being meaningful or not. Therefore, you shouldn't be subtracting anything from total.

Kinshook · Answer

In how many ways can a 4-letter word be formed from the letters ABCDEFIO such that the word contains 2 vowels and 2 consonants?

The letters cannot be repeated, and, the words can have no dictionary meaning.

Vowels - A-1, E-1, I-1, O-1
Consonants - B-1, C-1, D-1, F-1

The number of ways can a 4-letter word be formed from the letters ABCDEFIO such that the word contains 2 vowels and 2 consonants = Number of ways to select 2 vowels out of 4 * Number of ways to select 2 consonants out of 4* Arranging 4 letter to form a word = 4C2*4C2*4! = 6*6*24 = 864

IMO D

rahumangal · Answer

i think the answer is not considering what is given in the question stimulus ..i.e the words cannot have dictionary meaning....and so we should remove those words
for eg the word- ACID...and many more.
this is not typical of gmat question
i think question needs a rework

Bunuel · Answer

You misunderstood that part. “Words can have no dictionary meaning” just means they don’t  need  to be real words. It doesn’t mean you exclude real ones like ACID.

In how many ways can a 4-letter word be formed from the letters ABCDEF

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GMAT Problem Solving (PS) Questions

In how many ways can a 4-letter word be formed from the letters ABCDEF

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