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In how many ways all the letters of the word WEAPONS can be arranged

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In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 05 Mar 2019, 20:42
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Question Stats:

23% (02:07) correct 77% (01:47) wrong based on 82 sessions

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In how many ways all the letters of the word WEAPONS can be arranged such that in any of the arrangements, no two vowels will be together?

    A. \(^5C_3 * 3!\)

    B. \(3! * 4!\)

    C. \(^5C_3 * 4!\)

    D. \(7! – 5! * 3!\)

    E. \(^5C_3 * 3! * 4!\)

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In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post Updated on: 08 Mar 2019, 02:16
WEAPONS = 7! ways total arrangement
vowels = EAO 3! ways and consonants WPNS = 4! ways

these vowels can be arranged in 5c3 ways between 4 WPNS

5c3*3!*4!
IMO E


EgmatQuantExpert wrote:
In how many ways all the letters of the word WEAPONS can be arranged such that in any of the arrangements, no two vowels will be together?

    A. \(^5C_3 * 3!\)

    B. \(3! * 4!\)

    C. \(^5C_3 * 4!\)

    D. \(7! – 5! * 3!\)

    E. \(^5C_3 * 3! * 4!\)

Image

Originally posted by Archit3110 on 06 Mar 2019, 00:57.
Last edited by Archit3110 on 08 Mar 2019, 02:16, edited 1 time in total.
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Re: In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 06 Mar 2019, 18:23
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is ans E

arrange consonants as _W_P_N_S_ consonants E,A,O can be placed in any 5 blanks so 5c3

consonants can be arranged within itself in 3!
vowels can be arranged in 4! ways

so 5c3*4!*3!
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Re: In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 07 Mar 2019, 21:42

Solution


Given:
In this question, we are given
    • A specific word is given: WEAPONS, of which all the letters can be arranged to form different words

To find:
We need to determine
    • How many such arrangements are possible, where no two vowels will be together

Approach and Working:
In the given word “WEAPONS”, there are 4 consonants: W, P, N, S and 3 vowels: E, A, O.
    • When we say that no two vowels are together, it means there will be at least one consonant between any two vowels.

As there are 4 consonants, let us first put them, as shown in the following figure:

Image

Now, for the 3 vowels, there will be actually 5 places where any of those vowels can be placed.

Image

    • Out of the 5 possible places, 3 will be chosen for the vowels in \(^5C_3\) ways.
    • Also, the vowels can rearrange themselves in 3! ways and the consonants can rearrange themselves in 4! ways.
    • Therefore, the total number of possible arrangements = \(^5C_3 * 3! * 4!\)

Hence, the correct answer is option E.

Answer: E
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Re: In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 07 Mar 2019, 21:45
Archit3110 wrote:
WEAPONS = 7! ways total arrangement
vowels = EAO WPNS = 5! ways and total vowels = 3 ;3! ways

7!-5!*3!
IMO D


Hey Archit3110,
In your calculation, you missed those cases where two vowels can be together. You only removed the cases where all 3 vowels are together.
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Re: In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 07 Mar 2019, 22:57
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There are 7 letters in the word WEAPONS with 3 vowels and 4 consonant of the 5 possible places, 3 will be chosen for the vowels in 5C35C3 ways.the vowels can rearrange themselves in 3! ways and the consonants can rearrange themselves in 4! ways.. Hence, number of arrangements of letters such that all vowels are all together is 5C3∗3!∗4! so the correct answer is E
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Re: In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 08 Mar 2019, 00:55
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EgmatQuantExpert wrote:
Archit3110 wrote:
WEAPONS = 7! ways total arrangement
vowels = EAO WPNS = 5! ways and total vowels = 3 ;3! ways

7!-5!*3!
IMO D


Hey,
In your calculation, you missed those cases where two vowels can be together. You only removed the cases where all 3 vowels are together.




And what is the best way to do it with that approach? (removing csaes from the total)
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Re: In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 08 Mar 2019, 02:13
EgmatQuantExpert wrote:
Archit3110 wrote:
WEAPONS = 7! ways total arrangement
vowels = EAO WPNS = 5! ways and total vowels = 3 ;3! ways

7!-5!*3!
IMO D


Hey Archit3110,
In your calculation, you missed those cases where two vowels can be together. You only removed the cases where all 3 vowels are together.


EgmatQuantExpert
yeah i made an error.. misread question.. edited thanks..
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In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 09 Mar 2019, 07:30
When we insert 1 vowel in the middle of each consonant and 1 vowel each at the corner, in total, it would make 9 spaces, however WEAPONS has only 7 spaces.
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In how many ways all the letters of the word WEAPONS can be arranged  [#permalink]

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New post 02 Jun 2020, 06:27
Below you find another approach, which I used to solve this question:

WEAPONS = 3 Vowels & 4 Not Vowels
No repeating letter => Total arrangement possibilities = 7!

What is the number of arrangement in which all three Vowels are together?
5!*3! = 6!
(3! refers to the arrangements between the Vowels as AEO, OEA...)

What is the number of arrangement in which two Vowels are together?
3*(6!*2!-6!) + 6! = 4*6!

6!*2! = arrangements in which two Vowels are together; However, we have already considered the arrangements with three Vowels => need to deduct by 6!)
This whole package is multiplied by 3, because we have 3 pairs of two vowels: AO, AE, OE. Finally we add 6! to compensate extra reductions.
For example for pair EA, EOA is additionally deducted. This is the case for 1/3*3*(6!*2!-6!)=6! arrangements


Final Calculation:
7! - 4*6! - 6! = 7*6!-5*6! = 2*6! = 5C3*3!*4!

E.
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In how many ways all the letters of the word WEAPONS can be arranged   [#permalink] 02 Jun 2020, 06:27

In how many ways all the letters of the word WEAPONS can be arranged

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