The number of words, with or without meaning, that can be formed using all the letters of the word ASSASSINATION so that the vowels occur together, is _____.

The word ASSASSINATION has 13 letters: A(3), S(4), I(2), N(2), T(1), O(1). The vowels are A, A, A, I, I, O (6 vowels), and the consonants are S, S, S, S, N, N, T (7 consonants). Since the vowels must occur together, we treat all 6 vowels as a single block.

We then arrange this block along with the 7 consonants, giving 8 items in total with repetitions S(4), N(2), and T(1). The number of arrangements is $$ \text{Arrangements} = \frac{8!}{4! \times 2! \times 1!} = \frac{40320}{48} = 840 $$. Next, arranging the vowels within their block, where there are A(3), I(2), and O(1), yields $$ \text{Arrangements} = \frac{6!}{3! \times 2! \times 1!} = \frac{720}{12} = 60 $$. Therefore, the total number of arrangements is $$840 \times 60 = 50400$$.