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 with A appearing 3 times, S appearing 4 times, I appearing 2 times, N appearing 2 times, T once, and O once.

There are 6 vowels in total (A(3), I(2), O(1)) and 7 consonants (S(4), N(2), T(1)).

Since all the vowels must occur together, we treat these 6 vowels as a single block. This gives us 7 consonants plus 1 vowel block, making 8 units to arrange.

The 8 units consist of S, S, S, S, N, N, T and the vowel block. Using the multinomial formula to account for repeated consonants, their number of arrangements is:

$$\frac{8!}{4! \times 2!} = \frac{40320}{24 \times 2} = \frac{40320}{48} = 840$$

Substituting the 6 vowels A, A, A, I, I and O into the block, they can be arranged among themselves in

$$\frac{6!}{3! \times 2!} = \frac{720}{6 \times 2} = \frac{720}{12} = 60$$

Using the rule of product, the total number of arrangements with all vowels together is

$$840 \times 60 = 50400$$

This gives the final answer as 50400.