(i) The word HAPPINESS consists of 9 letters where P appears twice and S appears twice. The number of different arrangements is given by:

\(\frac{9!}{2! \times 2!} = \frac{362880}{4} = 90720\)

(ii) To find the probability that the 3 vowels (A, E, I) are not all next to each other, first calculate the number of arrangements where they are together. Treat the 3 vowels as a single unit, so we have 7 units to arrange: (AEI), H, P, P, N, S, S.

The number of arrangements of these 7 units is:

\(\frac{7!}{2! \times 2!} = \frac{5040}{4} = 1260\)

Within the unit (AEI), the vowels can be arranged in \(3!\) ways:

\(3! = 6\)

Thus, the total number of arrangements where the vowels are together is:

\(1260 \times 6 = 7560\)

The probability that the vowels are not all together is:

\(1 - \frac{7560}{90720} = \frac{83160}{90720} = 0.917 \left(=\frac{11}{12}\right)\)