In how many ways can the letters of the word COMPUTER be arranged so that all the vowels always stay together?

COMPUTER has 8 distinct letters, of which O, U, E are vowels. Tie the 3 vowels into one block: that leaves 5 consonants + 1 block = 6 units, arranged in 6! = 720 ways. The vowels shuffle inside the block in 3! = 6 ways. Total = $720\times 6=4320$.