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In how many ways can the letters of the word ABACUS

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by BTGmoderatorDC » Wed Nov 29, 2017 3:25 pm
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

Can some experts show me the best solution in this problem?

OA D
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by GMATWisdom » Wed Nov 29, 2017 5:03 pm
lheiannie07 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

Can some experts show me the best solution in this problem?

OA D
Permutation of N objects that have N1 identical objects of type 1 and N2 identical objects of type 2 and Nk identical objects of type k is

N!/(N1!*N2!..Nk!)

We have 3 vowels and 2 of then are identical (2 As), so the number of ways to arrange them are

3!/2!

Since we have to keep the vowels together, let us consider then as 1 group and call it X.

We then have to arrange the following alphabets.
XBCS

These can be arranged in 4! ways.

Hence the total number of ways in which these alphabets can be arranged is

4!*3!/2!

Hence D.
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by Scott@TargetTestPrep » Mon Oct 07, 2019 6:54 pm
BTGmoderatorDC 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

Can some experts show me the best solution in this problem?

OA D

We need to arrange [A-A-U] - B - C - S

Since A-A-U is considered one letter, the total arrangement of the 4 items can be arranged in 4! ways.

A-A-U can be arranged in 3!/2! = 3 ways.

Thus, the total number of ways to arrange the letters with the vowels together is 4! x 3!/2! ways.

Answer: D

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