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In how many ways can the letters of the word ACUMEN be rearranged such

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In how many ways can the letters of the word ACUMEN be rearranged such [#permalink]

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In how many ways can the letters of the word ACUMEN be rearranged such that the vowels always appear together?

A. 3!3!

B. \(\frac{6!}{2!}\)\(\)

C.\(\frac{4!3!}{2!}\)

D. 4!3!

E. \(\frac{3!3!}{2!}\)
[Reveal] Spoiler: OA

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Re: In how many ways can the letters of the word ACUMEN be rearranged such [#permalink]

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fitzpratik wrote:
In how many ways can the letters of the word ACUMEN be rearranged such that the vowels always appear together?

A. 3!3!

B. \(\frac{6!}{2!}\)\(\)

C. \(\frac{4!3!}{2!}\)

D. 4!3!

E. \(\frac{3!3!}{2!}\)


Consider the vowels as one unit: {AUE}. We'll have total of 4 units: {AUE}{C}{M}{N}, which can be arranged in 4! ways. A, U and E within the unit can be arranged in 3! ways, so the final answer is 4!*3!.

Answer: D.
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Re: In how many ways can the letters of the word ACUMEN be rearranged such [#permalink]

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New post 01 Sep 2017, 03:40
AEU together lets say it is (AEU) CMN remains
Now (AEU) is one compound alphabet and CMN are other 3
so total is 4! but AEU can rotate within brackets so 3!
total is 4!3!

Ans is D
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Re: In how many ways can the letters of the word ACUMEN be rearranged such [#permalink]

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New post 07 Sep 2017, 07:11
fitzpratik wrote:
In how many ways can the letters of the word ACUMEN be rearranged such that the vowels always appear together?

A. 3!3!

B. \(\frac{6!}{2!}\)\(\)

C.\(\frac{4!3!}{2!}\)

D. 4!3!

E. \(\frac{3!3!}{2!}\)


Since the vowels must be together we can arrange the letters as follows, treating the vowels as a single unit:

[A-U-E] - C - M - N

So we see there are 4! ways to arrange the above group of letters if we consider A-U-E as a single unit. Furthermore, there are also 3! ways to arrange A-U-E, so the word can be arranged in (4!)(3!) ways with the vowels always appearing together.

Answer: D
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Re: In how many ways can the letters of the word ACUMEN be rearranged such   [#permalink] 07 Sep 2017, 07:11
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