This sample GMAT Math question is a 600 level combinatorics problem solving question. The concept tested is to find the number of ways the letters of word can be rearranged after factoring in constraints that a certain type of letters should be grouped together. An elementary permutation question on rearragements and reordering objects.

Question 1: In how many ways can the letters of the word ABACUS be rearranged such that the vowels always appear together?

- \\frac{6!}{2!}\\)
- 3! × 3!
- \\frac{4!}{2!}\\)
- \\frac{4! \times 3!}{2!}\\)
- \\frac{3! \times 3!}{2!}\\)

@ INR

ABACUS is a 6 letter word in which 3 of the letters are vowels, viz., A, A, and U.

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.

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

Hence, the total number of rearrangements of the word ABACUS in which the vowels appear together is \\frac{4! \times 3!}{2!}\\)

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