In how many different ways can the letters of the word LEADING be arranged so that the vowels always occupy consecutive positions?

LEADING has 7 distinct letters with vowels E, A, I. Bind the vowels into one block: 4 consonants + 1 block = 5 units, arranged in 5! = 120 ways, and the vowels rearrange internally in 3! = 6 ways. Total = $120\times 6=720$.