In how many different ways, can the letters of the word ASSOCIATION be arranged, so that the vowels always come together?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 32400
When some letters must stay together, they are tied into one block and the block is arranged with the other letters; then the letters inside the block are arranged. Repeated letters are allowed for by dividing by the factorial of each repeat.
ASSOCIATION has 11 letters: vowels A, A, O, O, I, I (6) and consonants S, S, C, T, N (5)
The vowel block and the 5 consonants make 6 units, with S twice: \(\displaystyle \frac{6!}{2!}=\frac{720}{2}=360\) arrangements
The 6 vowels inside the block, with A, O and I each twice: \(\displaystyle \frac{6!}{2!\times2!\times2!}=\frac{720}{8}=90\) arrangements
Total \(=360\times90=32400\)