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# In how many ways can the letters of the word ABACUS be rearranged

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

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14 Mar 2016, 08:19
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In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

A. 6!/2!
B. 3!*3!
C. 4!/2!
D. 4! *3!/2!
E. 3!*3!/2

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

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14 Mar 2016, 11:54
In the word ABACUS , there are 3 vowels - 2 A's and U
Number of ways the letters of word ABACUS be rearranged such that the vowels always appear together
= (4! * 3! )/2!
We can consider the the 3 vowels as a single unit and there are 3 ways to arrange them . But since 2 elements of vowel group are identical we divide by 2! .
The entire vowel group is considered as a single group .

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

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02 Jul 2017, 10:03
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ABACUS = AAU BCS = Let's say AAU is altogether one entity
so we have 4 entities = (AAU)(BCS)
= so 4 entities can be arranged in 4! ways and within them ( the group - AAU can be arranged in $$\frac{3!}{2!}$$) ways

so total number of ways = 4! * $$\frac{3!}{2!}$$ = 24 * $$\frac{6}{2}$$ = 24 * 3 = 72
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Re: In how many ways can the letters of the word ABACUS be rearranged  [#permalink]

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02 Jul 2017, 10:08
Since the vowels, must always appear together(you combine the vowels as 1 unit)
Lets call this unit X. Now we have 4 letters, X B C S which can be arranged in 4!(24) ways

Now coming to the ways the vowels can be arranged, since there are two A's and 1 U
They can be arranged AUA, UAA and AAU(3 ways)

Total ways of arranging the alphabets of ABACUS
such that vowels occur togther : 24*3 = 72(Option D)
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Re: In how many ways can the letters of the word ABACUS be rearranged  [#permalink]

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03 Jul 2017, 21:33
Bunuel wrote:
jokschmer wrote:
In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

a) 144

b) 12

c) 36

d) 72

e) 81

Merging topics. Please refer to the solutions above.

Considering vowels as a single unit. It would be like {AAU}{B}{C}{S} so number of arrangements for this would be 4! = 4*3*2*1 = 24 AND {AAU} consists of 3 letters so it can be arranged in 3! ways = 3*2*1 = 6 ways. So, total number of arrangements would be 4! * 3! = 24 * 6 = 144 ways.

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

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03 Jul 2017, 22:47
reachskishore wrote:
Bunuel wrote:
jokschmer wrote:
In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

a) 144

b) 12

c) 36

d) 72

e) 81

Merging topics. Please refer to the solutions above.
3!/2!

Considering vowels as a single unit. It would be like {AAU}{B}{C}{S} so number of arrangements for this would be 4! = 4*3*2*1 = 24 AND {AAU} consists of 3 letters so it can be arranged in 3! ways = 3*2*1 = 6 ways. So, total number of arrangements would be 4! * 3! = 24 * 6 = 144 ways.

AAU has two identical letters, hence the ways of arranging them will be will be $$\frac{3!}{2!}$$ or 3 ways
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Re: In how many ways can the letters of the word ABACUS be rearranged  [#permalink]

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05 Apr 2018, 11:15
Bunuel wrote:
In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

A. 6!/2!
B. 3!*3!
C. 4!/2!
D. 4! *3!/2!
E. 3!*3!/2

We can arrange the letters as follows:

[A-A-U] - B - C - S

Thinking of [A-A-U] as a single element, [A-A-U] - B - C - S can be arranged in 4! ways.

We must also consider that [A-A-U] can be arranged in 3!/2! ways (by the formula for permutations with indistinguishable objects).

Thus, the total number of ways is 4! * 3!/2!.

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Re: In how many ways can the letters of the word ABACUS be rearranged &nbs [#permalink] 05 Apr 2018, 11:15
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