Choose the correct answer. In how many different ways can the letters of the word ' HARDWARE' be arranged in such a way that the vowels always come together.

To solve this problem, we first consider the vowels in the word 'HARDWARE' as a single entity. The vowels are A, A, E.

Step 1: Treat the vowels (AAE) as one letter and the rest of the letters individually. This gives us H, R, D, W, R, (AAE). So, we have 6 items to arrange.

Step 2: The number of ways to arrange these 6 items (considering that R is repeated) is 6!/2! = 360 ways.

Step 3: However, within the (AAE) entity, these 3 vowels can be arranged among themselves in 3!/2! = 3 ways (since A is repeated).

Step 4: Therefore, the total number of ways the letters in 'HARDWARE' can be arranged such that the vowels always come together is 360 * 3 = 1080 ways.