In how many different ways can the letters of the word 'LEADING' be arranged so that the vowels always come together?

Problem restatementCount arrangements of 'LEADING' where the vowels form one adjacent block.

Given data

• Word: L, E, A, D, I, N, G (7 distinct letters).
• Vowels: E, A, I (3 vowels).
• Consonants: L, D, N, G (4 consonants).

Concept/ApproachBlock the vowels as [V]; arrange [V] with the 4 consonants (5 items), then permute vowels inside [V].

Step-by-step calculation Arrange 5 items: 5! = 120 Permute vowels (E, A, I): 3! = 6 Total arrangements = 120 × 6 = 720

Verification/AlternativeAll letters are distinct; no division by factorials for repeats is needed.

Final Answer720