# GMAT Math Practice : Rearranging Letters

Concept: Rearrangement of objects. Grouping objects based on constraints.

This sample GMAT Math question is a combinatorics problem solving question. The concept tested is to find the number of ways the letters of word can be rearranged after factoring in the constraint that certain category of letters should be grouped together. An elementary permutation question.

#### Question: In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

1. $\frac{6!}{2!}\\$
2. 3! * 3!
3. $\frac{4!}{2!}\\$
4. $\frac{4!*3!}{2!}\\$
5. $\frac{3!*3!}{2!}\\$

#### Group the vowels as one unit and rearrange

ABACUS is a 6 letter word with 3 of the letters being vowels.

Because the 3 vowels have to appear together, let us group the AAU as one unit.

There are 3 consonants in addition to one unit of vowels.

These 4 elements can be rearranged in 4! ways.

#### Rearrange the letters within the unit containing the vowels

The 3 vowels can rearrange among themselves in $\frac{3!}{2!}\\$ ways because "a" appears twice.

Hence, the total number of rearrangements in which the vowels appear together is $\frac{4!*3!}{2!}\\$

Choice D is the correct answer.

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