(i) The word REMEMBRANCE has 11 letters with repetitions: 3 E's and 2 M's. The number of different arrangements is given by:

\(\frac{11!}{3! \times 2!} = 1663200\)

(ii) For arrangements starting and ending with M, consider the arrangement as M _ _ _ _ _ _ _ _ M. We arrange the remaining 9 letters (REEMBRANC) with 3 E's:

\(\frac{9!}{3!} = 30240\)

(iii) To find arrangements where the 4 vowels (E, E, A, E) are not together, first calculate the arrangements where they are together. Treat the 4 vowels as a single unit, so we have 8 units to arrange (4 vowels + 7 other letters):

\(\frac{8!}{2!} \times \frac{4!}{3!} = 40320\)

Subtract this from the total arrangements:

\(1663200 - 40320 = 1622880\)