Multiple choice

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

  1. 8640

  2. 4320

  3. 2160

  4. 17280

  5. None of these

Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

ABUNDANCE has 8 letters with A repeated twice. Vowels: A, U, A, E (4 letters, with A repeated twice). Treat vowels as one group: we have 5 entities (4 consonants + 1 vowel group). Arrangements = 5! × (4!/2!) = 120 × 12 = 1440. Within vowels: arrange A, U, A, E with A repeated = 4!/2! = 12. Total = 1440. Wait, let me recalculate: consonants (B, N, N, C) - N repeats twice. Total arrangements: (5! × 4!)/(2! × 2!) = (120 × 24)/(2 × 2) = 2880/4 = 720. Hmm, that's not matching either. Let me reconsider: treat AAAE as one group, arrange with B, N, N, C (5 items, N repeats twice) = 5!/2! = 60. Within vowels: A, U, A, E with A repeats = 4!/2! = 12. Total = 60 × 12 = 720. Still not matching options. Let me check: ABUNDANCE letters = A, B, U, N, D, A, N, C, E (9 letters, A twice, N twice). Vowels: A, U, A, E. Consonants: B, N, D, N, C (5 letters, N twice). Total arrangements with vowels together: treat vowels as one block. We have 6 items (5 consonants + 1 vowel block) with N repeating twice among consonants. Arrangement = 6!/2! × 4!/2! = 360 × 12 = 4320. Option B is correct.