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

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

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New post Updated on: 10 Apr 2018, 01:31
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A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

41% (01:00) correct 59% (01:13) wrong based on 71 sessions

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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

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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Originally posted by EgmatQuantExpert on 04 Apr 2018, 06:47.
Last edited by EgmatQuantExpert on 10 Apr 2018, 01:31, edited 6 times in total.
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Re: In how many ways can a 4-letter word be formed from the letters ABCDEF  [#permalink]

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New post 04 Apr 2018, 07:11
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EgmatQuantExpert wrote:

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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Re: In how many ways can a 4-letter word be formed from the letters ABCDEF  [#permalink]

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New post 08 Apr 2018, 21:50
1

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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Ace GMAT quant
Articles and Question to reach Q51 | Question of the week

Must Read Articles
Number Properties – Even Odd | LCM GCD
Word Problems – Percentage 1 | Percentage 2 | Time and Work 1 | Time and Work 2 | Time, Speed and Distance 1 | Time, Speed and Distance 2
Advanced Topics- Permutation and Combination 1 | Permutation and Combination 2 | Permutation and Combination 3 | Probability
Geometry- Triangles 1 | Triangles 2 | Triangles 3 | Common Mistakes in Geometry
Algebra- Wavy line

Practice Questions
Number Properties 1 | Number Properties 2 | Algebra 1 | Geometry | Prime Numbers | Absolute value equations | Sets



| '4 out of Top 5' Instructors on gmatclub | 70 point improvement guarantee | www.e-gmat.com

Re: In how many ways can a 4-letter word be formed from the letters ABCDEF &nbs [#permalink] 08 Apr 2018, 21:50
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