How many different ways can the letters of the word "MATERIAL" be arranged so that all the vowels appear together?

Explanation

The word "MATERIAL" contains the vowels A, A, E, and I. Treating all vowels as a single block, we combine them into one unit. Along with the consonants M, T, R, and L, this gives us 5 units to arrange. These 5 units can be arranged in 5! (120) ways. Inside the vowel block, the 4 vowels can be arranged among themselves in 4! (24) ways, but since there are two identical A's, we divide by 2! to account for repetition. Therefore, the total number of arrangements is 5! × (4!/2!) = 120 × (24/2) = 1440.