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In how many ways can a 4-letter word be formed from the letters ABCDEF Updated on: 07 Nov 2024, 06:59
Originally posted by EgmatQuantExpert on 04 Apr 2018, 06:47.
Last edited by Bunuel on 07 Nov 2024, 06:59, edited 8 times in total.
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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


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D
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In how many ways can a 4-letter word be formed from the letters ABCDEF 04 Apr 2018, 07:11
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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.

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

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
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In how many ways can a 4-letter word be formed from the letters ABCDEF 08 Apr 2018, 21:50
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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
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In how many ways can a 4-letter word be formed from the letters ABCDEF 21 Aug 2018, 14:30
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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


while solving the question, the statement "the words can have no dictionary meaning" has not been accounted for.
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In how many ways can a 4-letter word be formed from the letters ABCDEF 21 Aug 2018, 23:14
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monish447
while solving the question, the statement "the words can have no dictionary meaning" has not been accounted for.

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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In how many ways can a 4-letter word be formed from the letters ABCDEF 22 Aug 2018, 02:57
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monish447
while solving the question, the statement "the words can have no dictionary meaning" has not been accounted for.

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. :-)




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.
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In how many ways can a 4-letter word be formed from the letters ABCDEF 22 Aug 2018, 03:28
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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.

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. :-)
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In how many ways can a 4-letter word be formed from the letters ABCDEF 31 Jan 2019, 11:24
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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
\(?\,\,:\,\,\# \,\,4\,\,{\rm{distinct}}\,\,{\rm{letters}}\,\,\,\left\{ \matrix{\\
\,2\,\,{\rm{vowels}} \hfill \cr \\
\,2\,\,{\rm{consonants}} \hfill \cr} \right.\)

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


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

Regards,
Fabio.
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In how many ways can a 4-letter word be formed from the letters ABCDEF 14 Jun 2019, 11:06
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C= BCDF
V= AEIO
4c2*4c2*4! = 864
IMOD

EgmatQuantExpert
Learn when to “Add” and “Multiply” in Permutation & Combination questions- Exercise Question #5

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.

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

Learn to use the Keyword Approach in Solving PnC question from the following article:

Learn when to “Add” and “Multiply” in Permutation & Combination questions


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In how many ways can a 4-letter word be formed from the letters ABCDEF 06 Nov 2022, 19:59
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Learn when to “Add” and “Multiply” in Permutation & Combination questions- Exercise Question #5

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.

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

‘ 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
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In how many ways can a 4-letter word be formed from the letters ABCDEF 07 Nov 2024, 05:59
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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
Why cant we use the approach P(4,2) [for vowels] * P(4,2) [for consonants]. Cause the order in selection should ideally matter while selecting the numbers,
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