In how many ways can the letters of the word ABACUS

BREAKING: Target Test Prep releases Brand New 2026 On Demand GMAT prep course

Redeem

This topic has expert replies

Post new topic Post Reply

Previous Topic Next Topic

BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pm; Followed by:23 members

In how many ways can the letters of the word ABACUS

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

Quote

Source: Beat The GMAT — Problem Solving | Wed Nov 29, 2017 3:25 pm

GMATWisdom: Master | Next Rank: 500 Posts; Posts: 100; Joined: Wed Nov 29, 2017 4:38 pm; Thanked: 14 times

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.

Quote

GMAT/MBA Expert

Scott@TargetTestPrep: GMAT Instructor; Posts: 8083; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

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

Scott Woodbury-Stewart
Founder and CEO
[email protected]