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In how many ways can the letters of the word ABACUS...

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by GMATGuruNY » Wed Mar 28, 2018 6:51 pm
BTGmoderatorLU 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!
Since the vowels must appear together, put them together in a BLOCK: [AAU].
Now count the number of ways to arrange the 4 elements [AAU], B, C and S.
The number of ways to arrange 4 distinct elements = 4!.

Now we must account for the number of ways that the vowels themselves can be arranged WITHIN the [AAU] block.
The vowels can be arranged as follows:
AAU, AUA, UAA.
Total ways = 3.

Multiplying the results above, we get:
4! * 3.
This is the value yielded by answer choice D:
(4! * 3!)/2! = 4! * 3.

The correct answer is D.
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by Jeff@TargetTestPrep » Thu Apr 05, 2018 10:19 am
BTGmoderatorLU 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!.

Answer: D

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